15858:
20755:
20285:
15081:
564:). The first element of a CIRC decoder is a relatively weak inner (32,28) Reed–Solomon code, shortened from a (255,251) code with 8-bit symbols. This code can correct up to 2 byte errors per 32-byte block. More importantly, it flags as erasures any uncorrectable blocks, i.e., blocks with more than 2 byte errors. The decoded 28-byte blocks, with erasure indications, are then spread by the deinterleaver to different blocks of the (28,24) outer code. Thanks to the deinterleaving, an erased 28-byte block from the inner code becomes a single erased byte in each of 28 outer code blocks. The outer code easily corrects this, since it can handle up to 4 such erasures per block.
10972:
20298:
19828:
10313:
15853:{\displaystyle {\begin{bmatrix}\Lambda _{3}S_{6}&x^{8}\\\Lambda _{2}S_{6}+\Lambda _{3}S_{5}&x^{7}\\\Lambda _{1}S_{6}+\Lambda _{2}S_{5}+\Lambda _{3}S_{4}&x^{6}\\S_{6}+\Lambda _{1}S_{5}+\Lambda _{2}S_{4}+\Lambda _{3}S_{3}&x^{5}\\S_{5}+\Lambda _{1}S_{4}+\Lambda _{2}S_{3}+\Lambda _{3}S_{2}&x^{4}\\S_{4}+\Lambda _{1}S_{3}+\Lambda _{2}S_{2}+\Lambda _{3}S_{1}&x^{3}\\S_{3}+\Lambda _{1}S_{2}+\Lambda _{2}S_{1}&x^{2}\\S_{2}+\Lambda _{1}S_{1}&x\\S_{1}\end{bmatrix}}={\begin{bmatrix}Q_{2}x^{8}\\Q_{1}x^{7}\\Q_{0}x^{6}\\0\\0\\0\\\Omega _{2}x^{2}\\\Omega _{1}x\\\Omega _{0}\end{bmatrix}}}
20750:{\displaystyle {\begin{bmatrix}001&000&000&000&000&000&000\\000&001&000&000&000&000&000\\000&000&001&000&000&000&000\\000&000&000&001&000&000&000\\000&000&000&000&001&000&000\\000&000&000&000&000&001&000\\000&000&000&000&000&000&001\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q_{0}\\q_{1}\\q_{2}\\q_{3}\\q_{4}\end{bmatrix}}={\begin{bmatrix}006\\924\\006\\007\\009\\916\\003\end{bmatrix}}}
20280:{\displaystyle {\begin{bmatrix}001&000&928&000&000&000&000\\006&006&928&928&928&928&928\\123&246&928&927&925&921&913\\456&439&928&926&920&902&848\\057&228&928&925&913&865&673\\086&430&928&924&904&804&304\\121&726&928&923&893&713&562\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q_{0}\\q_{1}\\q_{2}\\q_{3}\\q_{4}\end{bmatrix}}={\begin{bmatrix}000\\923\\437\\541\\017\\637\\289\end{bmatrix}}}
12568:
9704:
10967:{\displaystyle {\begin{aligned}&Y_{k}X_{k}^{j+\nu }\Lambda (X_{k}^{-1})=0.\\&Y_{k}X_{k}^{j+\nu }\left(1+\Lambda _{1}X_{k}^{-1}+\Lambda _{2}X_{k}^{-2}+\cdots +\Lambda _{\nu }X_{k}^{-\nu }\right)=0.\\&Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu }X_{k}^{-1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu }X_{k}^{-2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j+\nu }X_{k}^{-\nu }=0.\\&Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j}=0.\end{aligned}}}
12214:
3087:
9335:
2735:
19247:
potential polynomials, until a sufficient number of matching polynomials are produced to reasonably eliminate any errors in the received codeword. Once a polynomial is determined, then any errors in the codeword can be corrected, by recalculating the corresponding codeword values. Unfortunately, in all but the simplest of cases, there are too many subsets, so the algorithm is impractical. The number of subsets is the
14983:
445:
11879:
11535:
708:
12563:{\displaystyle {\begin{bmatrix}S_{1}&S_{2}&\cdots &S_{\nu }\\S_{2}&S_{3}&\cdots &S_{\nu +1}\\\vdots &\vdots &\ddots &\vdots \\S_{\nu }&S_{\nu +1}&\cdots &S_{2\nu -1}\end{bmatrix}}{\begin{bmatrix}\Lambda _{\nu }\\\Lambda _{\nu -1}\\\vdots \\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}-S_{\nu +1}\\-S_{\nu +2}\\\vdots \\-S_{\nu +\nu }\end{bmatrix}}}
17114:
9699:{\displaystyle {\begin{bmatrix}X_{1}^{1}&X_{2}^{1}&\cdots &X_{\nu }^{1}\\X_{1}^{2}&X_{2}^{2}&\cdots &X_{\nu }^{2}\\\vdots &\vdots &\ddots &\vdots \\X_{1}^{n-k}&X_{2}^{n-k}&\cdots &X_{\nu }^{n-k}\\\end{bmatrix}}{\begin{bmatrix}Y_{1}\\Y_{2}\\\vdots \\Y_{\nu }\end{bmatrix}}={\begin{bmatrix}S_{1}\\S_{2}\\\vdots \\S_{n-k}\end{bmatrix}}}
7196:). The "missing" bits in a shortened code need to be filled by either zeros or ones, depending on whether the data is complemented or not. (To put it another way, if the symbols are inverted, then the zero-fill needs to be inverted to a one-fill.) For this reason it is mandatory that the sense of the data (i.e., true or complemented) be resolved before Reed–Solomon decoding.
8871:
415:(1961). By 1963 (or possibly earlier), J. J. Stone (and others) recognized that Reed Solomon codes could use the BCH scheme of using a fixed generator polynomial, making such codes a special class of BCH codes, but Reed Solomon codes based on the original encoding scheme, are not a class of BCH codes, and depending on the set of evaluation points, they are not even
4942:
14629:
11562:
11218:
7181:. This is because it does not matter to the code how many bits in a symbol are in error — if multiple bits in a symbol are corrupted it only counts as a single error. Conversely, if a data stream is not characterized by error bursts or drop-outs but by random single bit errors, a Reed–Solomon code is usually a poor choice compared to a binary code.
13945:
3082:{\displaystyle C(m)=Am={\begin{bmatrix}1&a_{0}&a_{0}^{2}&\dots &a_{0}^{k-1}\\1&a_{1}&a_{1}^{2}&\dots &a_{1}^{k-1}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&a_{n-1}&a_{n-1}^{2}&\dots &a_{n-1}^{k-1}\end{bmatrix}}{\begin{bmatrix}m_{0}\\m_{1}\\\vdots \\m_{k-1}\end{bmatrix}}}
16835:
386:). The original encoding scheme described in the Reed & Solomon article used a variable polynomial based on the message to be encoded where only a fixed set of values (evaluation points) to be encoded are known to encoder and decoder. The original theoretical decoder generated potential polynomials based on subsets of
1576:
11211:
8597:
14073:
4739:
4704:
7685:
Designers are not required to use the "natural" sizes of Reed–Solomon code blocks. A technique known as "shortening" can produce a smaller code of any desired size from a larger code. For example, the widely used (255,223) code can be converted to a (160,128) code by padding the unused portion of the
8974:
The advantage of looking at the syndromes is that the message polynomial drops out. In other words, the syndromes only relate to the error, and are unaffected by the actual contents of the message being transmitted. If the syndromes are all zero, the algorithm stops here and reports that the message
398:
like scheme based on a fixed polynomial known to both encoder and decoder, but later, practical decoders based on the original scheme were developed, although slower than the BCH schemes. The result of this is that there are two main types of Reed
Solomon codes, ones that use the original encoding
19180:
The decoders described in this section use the Reed
Solomon original view of a codeword as a sequence of polynomial values where the polynomial is based on the message to be encoded. The same set of fixed values are used by the encoder and decoder, and the decoder recovers the encoding polynomial
19246:
and which encoding method was used to generate the codeword's sequence of values. The original message, the polynomial, and any errors are unknown. A decoding procedure could use a method like
Lagrange interpolation on various subsets of n codeword values taken k at a time to repeatedly produce
13793:
567:
The result is a CIRC that can completely correct error bursts up to 4000 bits, or about 2.5 mm on the disc surface. This code is so strong that most CD playback errors are almost certainly caused by tracking errors that cause the laser to jump track, not by uncorrectable error bursts.
6509:
6661:
14978:{\displaystyle {\begin{aligned}S(x)&=S_{t}x^{t-1}+S_{t-1}x^{t-2}+\cdots +S_{2}x+S_{1}\\\Lambda (x)&=\Lambda _{e}x^{e}+\Lambda _{e-1}x^{e-1}+\cdots +\Lambda _{1}x+1\\\Omega (x)&=\Omega _{e}x^{e}+\Omega _{e-1}x^{e-1}+\cdots +\Omega _{1}x+\Omega _{0}\end{aligned}}}
9992:
11874:{\displaystyle \left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu }\right)+\Lambda _{1}\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu -1}\right)+\Lambda _{2}\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu -2}\right)+\cdots +\Lambda _{\nu }\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j}\right)=0}
11530:{\displaystyle \left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu }\right)+\left(\sum _{k=1}^{\nu }\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}\right)+\left(\sum _{k=1}^{\nu }\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}\right)+\cdots +\left(\sum _{k=1}^{\nu }\Lambda _{\nu }Y_{k}X_{k}^{j}\right)=0}
6316:
13651:
1422:
19822:
10987:
7719:
Daniel
Gorenstein and Neal Zierler developed a decoder that was described in a MIT Lincoln Laboratory report by Zierler in January 1960 and later in a paper in June 1961. The Gorenstein–Zierler decoder and the related work on BCH codes are described in a book
7203:
or not depends on subtle details of the construction. In the original view of Reed and
Solomon, where the codewords are the values of a polynomial, one can choose the sequence of evaluation points in such a way as to make the code cyclic. In particular, if
17301:
The algebraic decoding methods described above are hard-decision methods, which means that for every symbol a hard decision is made about its value. For example, a decoder could associate with each symbol an additional value corresponding to the channel
13956:
22375:
17228:
at MIT published "Improved
Decoding of Reed–Solomon and Algebraic-Geometry Codes" introducing an algorithm that allowed for the correction of errors beyond half the minimum distance of the code. It applies to Reed–Solomon codes and more generally to
17109:{\displaystyle {\begin{aligned}E_{0}&=-{\frac {1}{\Lambda _{v}}}(E_{v}+\Lambda _{1}E_{v-1}+\cdots +\Lambda _{v-1}E_{1})\\E_{j}&=-(\Lambda _{1}E_{j-1}+\Lambda _{2}E_{j-2}+\cdots +\Lambda _{v}E_{j-v})&{\text{for }}t<j<n\end{aligned}}}
12019:
12189:
4471:
3951:
3576:
2602:
410:
and Neal
Zierler was described in an MIT Lincoln Laboratory report by Zierler in January 1960 and later in a paper in June 1961. The Gorenstein–Zierler decoder and the related work on BCH codes are described in a book Error Correcting Codes by
17326:
presented a polynomial-time soft-decision algebraic list-decoding algorithm for Reed–Solomon codes, which was based upon the work by Sudan and
Guruswami. In 2016, Steven J. Franke and Joseph H. Taylor published a novel soft-decision decoder.
4120:
However, Lagrange interpolation performs the same conversion without the constraint on the set of evaluation points or the requirement of an error free set of message values and is used for systematic encoding, and in one of the steps of the
9734:) are already known by some other method (for example, in an FM transmission, the sections where the bitstream was unclear or overcome with interference are probabilistically determinable from frequency analysis). In this scenario, up to
6077:
394:(encoded message length) values of a received message, choosing the most popular polynomial as the correct one, which was impractical for all but the simplest of cases. This was initially resolved by changing the original scheme to a
13321:
8866:{\displaystyle {\begin{aligned}S_{j}&=r(\alpha ^{j})=s(\alpha ^{j})+e(\alpha ^{j})=0+e(\alpha ^{j})\\&=e(\alpha ^{j})\\&=\sum _{k=1}^{\nu }e_{i_{k}}\left(\alpha ^{j}\right)^{i_{k}},\quad j=1,2,\ldots ,n-k\end{aligned}}}
9288:
6334:
13003:
4116:
16705:
14371:
13475:
4937:{\displaystyle \mathbf {C} =\left\{\left(s_{1},s_{2},\dots ,s_{n}\right)\;{\Big |}\;s(a)=\sum _{i=1}^{n}s_{i}a^{i}{\text{ is a polynomial that has at least the roots }}\alpha ^{1},\alpha ^{2},\dots ,\alpha ^{n-k}\right\}.}
12582:
and sets up the linear system for that value. If the equations can be solved (i.e., the matrix determinant is nonzero), then that trial value is the number of errors. If the linear system cannot be solved, then the trial
13150:
8131:
13789:
6514:
10318:
3324:
19375:
was developed as a decoder that is able to recover the original message polynomial as well as an error "locator" polynomial that produces zeroes for the input values that correspond to errors, with time complexity
9838:
14634:
22381:
13940:{\displaystyle {\begin{bmatrix}732&637\\637&762\end{bmatrix}}{\begin{bmatrix}\Lambda _{2}\\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}-762\\-925\end{bmatrix}}={\begin{bmatrix}167\\004\end{bmatrix}}}
13488:
7689:
The QR code, Ver 3 (29×29) uses interleaved blocks. The message has 26 data bytes and is encoded using two Reed-Solomon code blocks. Each block is a (255,233) Reed
Solomon code shortened to a (35,13) code.
19604:
5529:
8482:
7495:
614:
use Reed–Solomon error correction to allow correct reading even if a portion of the bar code is damaged. When the bar code scanner cannot recognize a bar code symbol, it will treat it as an erasure.
9084:
8000:
2270:
14236:
9162:
7603:
7811:
7192:
somewhere along the line, the decoders will still operate. The result will be the inversion of the original data. However, the Reed–Solomon code loses its transparency when the code is shortened (
1856:
15066:
8212:
16840:
8602:
11886:
5641:
3686:
12059:
2153:
20958:
19329:
8367:
7898:
5844:
21719:
10156:
8583:
3809:
3434:
2460:
12612:
are the reciprocals of those roots. The order of coefficients of the error location polynomial can be reversed, in which case the roots of that reversed polynomial are the error locators
6238:
erroneous symbols, i.e., it can correct half as many errors as there are redundant symbols added to the block. Sometimes error locations are known in advance (e.g., "side information" in
6716:
6822:
19591:
336:
erasures at locations that are known and provided to the algorithm, or it can detect and correct combinations of errors and erasures. Reed–Solomon codes are also suitable as multiple-
7643:
10206:
5931:
7343:
512:
In 1996, variations of original scheme decoders called list decoders or soft decoders were developed by Madhu Sudan and others, and work continues on these types of decoders – see
7533:
3738:
3427:
2389:
10308:
1944:
21044:
2052:
1571:{\displaystyle \mathbf {C} ={\Bigl \{}\;{\bigl (}p(a_{1}),p(a_{2}),\dots ,p(a_{n}){\bigr )}\;{\Big |}\;p{\text{ is a polynomial over }}F{\text{ of degree }}<k\;{\Bigr \}}\,.}
1340:
6774:
4009:
13154:
10260:
8913:
2455:
1128:
11206:{\displaystyle \sum _{k=1}^{\nu }\left(Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j}\right)=0}
7705:
The decoders described in this section use the BCH view of a codeword as a sequence of coefficients. They use a fixed generator polynomial known to both encoder and decoder.
7172:
5683:
21466:
Authors in
Andrews et al. (2007), provide simulation results which show that for the same code rate (1/6) turbo codes outperform Reed-Solomon concatenated codes up to 2 dB (
8289:
12835:
7016:
6151:
5009:
1682:
10078:
5897:
1417:
1025:
8969:
7293:
12672:
12054:
10037:
4013:
19361:
17274:
6971:
4247:
2690:
21089:
19410:
16614:
14273:
11557:
5437:
21123:
13326:
12741:
7222:
6236:
4467:
2664:
1750:
20989:
14068:{\displaystyle {\begin{bmatrix}001&000\\000&001\end{bmatrix}}{\begin{bmatrix}\Lambda _{2}\\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}329\\821\end{bmatrix}}}
7066:
19244:
19217:
16830:
16803:
16761:
16734:
12637:
7379:
7122:
7096:
6912:
5352:
5320:
3804:
3765:
3351:
3207:
3180:
2317:
2180:
1982:
1890:
1196:
There are different encoding procedures for the Reed–Solomon code, and thus, there are different ways to describe the set of all codewords. In the original view of
1186:
13027:
8942:
8028:
5926:
5875:
5761:
5712:
5381:
5293:
5261:
5232:
5203:
5174:
5145:
5116:
5087:
5038:
4447:
4392:
4334:
4305:
4276:
4189:
4160:
14262:
is an alternate iterative procedure for finding the error locator polynomial. During each iteration, it calculates a discrepancy based on a current instance of Λ(
13655:
9758:
6196:
5557:
4732:
4699:{\displaystyle g(x)=\left(x-\alpha ^{i}\right)\left(x-\alpha ^{i+1}\right)\cdots \left(x-\alpha ^{i+n-k-1}\right)=g_{0}+g_{1}x+\cdots +g_{n-k-1}x^{n-k-1}+x^{n-k}}
4418:
4360:
4215:
1708:
1626:
1154:
9784:
1220:. In order to obtain a codeword of the Reed–Solomon code, the message symbols (each within the q-sized alphabet) are treated as the coefficients of a polynomial
19430:
12605:
found in the last step to build the error location polynomial. The roots of the error location polynomial can be found by exhaustive search. The error locators
10228:
9804:
7667:
7419:
7399:
7266:
7246:
7040:
6932:
6885:
6862:
6842:
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6290:
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5401:
5058:
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coefficients, with the constraint that in order for this to work, the set of evaluation points used to encode the message must be a set of increasing powers of
3602:
3371:
3232:
3227:
3141:
3113:
2730:
2710:
2622:
2409:
2337:
2290:
1600:
1282:
1262:
1238:
1218:
1088:
1068:
1048:
990:
967:
944:
7693:
The Delsarte–Goethals–Seidel theorem illustrates an example of an application of shortened Reed–Solomon codes. In parallel to shortening, a technique known as
9169:
672:
codewords. The code rate is generally set to 1/2 unless the channel's erasure likelihood can be adequately modelled and is seen to be less. In conclusion,
362:
view – with BCH view being the most common, as BCH view decoders are faster and require less working storage than original view decoders.
1094:. In the most useful parameterizations of the Reed–Solomon code, the block length is usually some constant multiple of the message length, that is, the
17279:
In 2023, building on three exciting works, coding theorists showed that Reed-Solomon codes defined over random evaluation points can actually achieve
6198:, the measure of redundancy in the block. If the locations of the error symbols are not known in advance, then a Reed–Solomon code can correct up to
22087:
21777:
1984:, and thus every message can be uniquely mapped to such a polynomial, there are different ways of doing this encoding. The original construction of
14595:
Another iterative method for calculating both the error locator polynomial and the error value polynomial is based on Sugiyama's adaptation of the
7686:
source block with 95 binary zeroes and not transmitting them. At the decoder, the same portion of the block is loaded locally with binary zeroes.
5442:
9814:
There is a linear recurrence relation that gives rise to a system of linear equations. Solving those equations identifies those error locations
6504:{\displaystyle P_{b}\approx {\frac {2^{m-1}}{2^{m}-1}}{\frac {1}{n}}\sum _{\ell =t+1}^{n}\ell {n \choose \ell }P_{s}^{\ell }(1-P_{s})^{n-\ell }}
19192:
described a theoretical decoder that corrected errors by finding the most popular message polynomial. The decoder only knows the set of values
8400:
9006:
7921:
4955:, in which each codeword contains the message as a prefix, and simply appends error correcting symbols as a suffix. Here, instead of sending
2185:
9091:
6253:) is able to correct twice as many erasures as errors, and any combination of errors and erasures can be corrected as long as the relation
692:
14990:
8136:
16591:) using discrete Fourier transform. Since the calculation for a discrete Fourier transform is the same as the calculation for syndromes,
514:
9806:
used thus far. This is why twice as many error correcting symbols need to be added as can be corrected without knowing their locations.
22468:
21841:
D. Gorenstein and N. Zierler, "A class of cyclic linear error-correcting codes in p^m symbols," J. SIAM, vol. 9, pp. 207–214, June 1961
22463:
8293:
7824:
5766:
22116:
19432:
is the number of values in a message. The recovered polynomial is then used to recover (recalculate as needed) the original message.
561:
6656:{\displaystyle P_{b}\approx {\frac {1}{m}}{\frac {1}{n}}\sum _{\ell =t+1}^{n}\ell {n \choose \ell }P_{s}^{\ell }(1-P_{s})^{n-\ell }}
740:
tend to produce errors in short bursts. Correcting these burst errors is a job best done by short or simplified Reed–Solomon codes.
734:, a practice that has since become very widespread in deep space and satellite (e.g., direct digital broadcasting) communications.
6153:
The Reed–Solomon code is optimal in the sense that the minimum distance has the maximum value possible for a linear code of size (
22563:
22538:
22354:
22200:
22038:
21874:
1346:, and the sequence of values is the corresponding codeword. Common choices for a set of evaluation points include {0, 1, 2, ...,
21682:
Hagenauer, J.; Offer, E.; Papke, L. (1994). "11. Matching Viterbi Decoders and Reed-Solomon Decoders in a Concatenated System".
9715:, because then the leftmost matrix would be known, and both sides of the equation could be multiplied by its inverse, yielding Y
21814:
9987:{\displaystyle \Lambda (x)=\prod _{k=1}^{\nu }(1-xX_{k})=1+\Lambda _{1}x^{1}+\Lambda _{2}x^{2}+\cdots +\Lambda _{\nu }x^{\nu }}
213:
19522:
7424:
22404:
22320:
22179:
21988:
21941:
21667:
21571:
14100:
with the log of the roots corresponding to the error locations (right to left, location 0 is the last term in the codeword).
17:
16763:
as syndromes (they're the same) and generate the error locator polynomial using the methods from any of the above decoders.
14149:
10161:
7538:
11883:
These summations are now equivalent to the syndrome values, which we know and can substitute in! This therefore reduces to
9325:
were known (see below), then the syndrome equations provide a linear system of equations that can easily be solved for the
7739:
3776:
1755:
680:, meaning that at least half of all the codewords sent must be received in order to reconstruct all of the codewords sent.
538:
Reed–Solomon coding is very widely used in mass storage systems to correct the burst errors associated with media defects.
17293:
errors) over linear size alphabets with high probability. However, this result is combinatorial rather than algorithmic.
12578:
directly but rather searches for it by trying successive values. The decoder first assumes the largest value for a trial
21561:
13646:{\displaystyle S_{1}=r(3^{1})=3\cdot 3^{6}+2\cdot 3^{5}+123\cdot 3^{4}+456\cdot 3^{3}+191\cdot 3^{2}+487\cdot 3+474=732}
7177:
The Reed–Solomon code properties discussed above make them especially well-suited to applications where errors occur in
5562:
3607:
19817:{\displaystyle b_{i}(e_{0}+e_{1}a_{i})-(q_{0}+q_{1}a_{i}+q_{2}a_{i}^{2}+q_{3}a_{i}^{3}+q_{4}a_{i}^{4})=-b_{i}a_{i}^{2}}
3974:
2081:
456:
20887:
22427:
21691:
19254:
17322:, has spurred interest in applying soft-decision decoding to conventional algebraic codes. In 2003, Ralf Koetter and
10083:
7669:. So choosing a sequence of primitive root powers as the evaluation points makes the original view Reed–Solomon code
22519:
FEC library in C by Phil Karn (aka KA9Q) includes Reed–Solomon codec, both arbitrary and optimized (223,255) version
8542:
8235:
21872:
Guruswami, V.; Sudan, M. (September 1999), "Improved decoding of Reed–Solomon codes and algebraic geometry codes",
1387:
9723:
In the variant of this algorithm where the locations of the errors are already known (when it is being used as an
6934:-bit value. The sender sends the data points as encoded blocks, and the number of symbols in the encoded block is
6250:
1947:
189:
2062:. The latter encoding procedure, while being slightly less efficient, has the advantage that it gives rise to a
464:
21422:
14382:
14259:
6779:
6666:
6172:
The error-correcting ability of a Reed–Solomon code is determined by its minimum distance, or equivalently, by
1892:
is the rate. This trade-off between the relative distance and the rate is asymptotically optimal since, by the
1200:, every codeword of the Reed–Solomon code is a sequence of function values of a polynomial of degree less than
922:
The Reed–Solomon code is actually a family of codes, where every code is characterised by three parameters: an
431:
169:
12205:
linear equations, not just one. This system of linear equations can therefore be solved for the coefficients Λ
22447:
21427:
19372:
17307:
7608:
6324:
1130:
is some constant, and furthermore, the block length is equal to or one less than the alphabet size, that is,
506:
495:
300:
elements called symbols. Reed–Solomon codes are able to detect and correct multiple symbol errors. By adding
571:
DVDs use a similar scheme, but with much larger blocks, a (208,192) inner code, and a (182,172) outer code.
21502:
Gorenstein, D.; Zierler, N. (June 1961). "A class of cyclic linear error-correcting codes in p^m symbols".
21442:
14596:
12014:{\displaystyle S_{j+\nu }+\Lambda _{1}S_{j+\nu -1}+\cdots +\Lambda _{\nu -1}S_{j+1}+\Lambda _{\nu }S_{j}=0}
7298:
743:
Modern versions of concatenated Reed–Solomon/Viterbi-decoded convolutional coding were and are used on the
522:
438:
14094:
The coefficients can be reversed to produce roots with positive exponents, but typically this isn't used:
12184:{\displaystyle S_{j}\Lambda _{\nu }+S_{j+1}\Lambda _{\nu -1}+\cdots +S_{j+\nu -1}\Lambda _{1}=-S_{j+\nu }}
7500:
6319:
Theoretical BER performance of the Reed-Solomon code (N=255, K=233, QPSK, AWGN). Step-like characteristic.
3691:
3380:
2342:
21959:"Randomly Punctured Reed-Solomon Codes Achieve the List Decoding Capacity over Polynomial-Size Alphabets"
21773:
10265:
7714:
1903:
21828:. Explains the Delsarte-Goethals-Seidel theorem as used in the context of the error correcting code for
20994:
2011:
1299:
6721:
719:
One significant application of Reed–Solomon coding was to encode the digital pictures sent back by the
480:
282:
206:
10233:
8876:
3946:{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}}
3571:{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}}
2597:{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}}
2414:
1100:
22076:
21654:(1994), "Reed–Solomon Codes and the Compact Disc", in Wicker, Stephen B.; Bhargava, Vijay K. (eds.),
7127:
5646:
312:
check symbols to the data, a Reed–Solomon code can detect (but not correct) any combination of up to
22543:
22052:
21888:
21447:
20873:
In 2002, an improved decoder was developed by Shuhong Gao, based on the extended Euclid algorithm.
17230:
6976:
6099:
4958:
1946:. Being a code that achieves this optimal trade-off, the Reed–Solomon code belongs to the class of
1631:
696:
22036:
Koetter, Ralf; Vardy, Alexander (2003). "Algebraic soft-decision decoding of Reed–Solomon codes".
19363:
code that can correct 3 errors, the naïve theoretical decoder would examine 359 billion subsets.
16444:
A discrete Fourier transform can be used for decoding. To avoid conflict with syndrome names, let
10042:
6111:
5880:
1400:
998:
545:. It was the first use of strong error correction coding in a mass-produced consumer product, and
21862:
Shu Lin and Daniel J. Costello Jr, "Error Control Coding" second edition, pp. 255–262, 1982, 2004
8947:
7271:
7225:
12642:
12026:
10007:
459:. The first commercial application in mass-produced consumer products appeared in 1982 with the
22568:
22047:
21883:
19334:
17315:
17240:
6937:
6328:
6072:{\displaystyle s(x)\equiv p(x)\cdot x^{t}-s_{r}(x)\equiv s_{r}(x)-s_{r}(x)\equiv 0\mod g(x)\,.}
4220:
3374:
2669:
752:
638:
379:
22007:
Randomly punctured Reed--Solomon codes achieve list-decoding capacity over linear-sized fields
21058:
19379:
11542:
9763:
The rest of the algorithm serves to locate the errors, and will require syndrome values up to
5406:
3146:
22394:
22144:
21095:
12726:
7207:
6242:
6201:
4452:
4449:
is defined as the polynomial whose roots are sequential powers of the Galois field primitive
2631:
1717:
230:
199:
22453:
22417:
20964:
7045:
970:
85:
20290:
19248:
19222:
19195:
17225:
16808:
16781:
16739:
16712:
13950:
12615:
7348:
7101:
7075:
6890:
5325:
5298:
4363:
3782:
3743:
3329:
3185:
3158:
2295:
2158:
1960:
1861:
1191:
1159:
923:
488:
437:
In 1975, another improved BCH scheme decoder was developed by Yasuo Sugiyama, based on the
22458:
22125:
8918:
5902:
5851:
5737:
5688:
5357:
5269:
5237:
5208:
5179:
5150:
5121:
5092:
5063:
5014:
4423:
4368:
4310:
4281:
4252:
4165:
4136:
947:
73:
8:
22488:
21918:. STOC 2023. New York, NY, USA: Association for Computing Machinery. pp. 1488–1501.
17318:
belief propagation decoding methods to achieve error-correction performance close to the
13316:{\displaystyle s(x)=p(x)\,x^{t}-s_{r}(x)=3x^{6}+2x^{5}+1x^{4}+382x^{3}+191x^{2}+487x+474}
9737:
6175:
5536:
4951:
The encoding procedure for the BCH view of Reed–Solomon codes can be modified to yield a
4711:
4397:
4339:
4194:
1687:
1605:
1133:
808:
712:
468:
173:
21718:
Andrews, K.S.; Divsalar, D.; Dolinar, S.; Hamkins, J.; Jones, C.R.; Pollara, F. (2007).
12201:
inclusive, and this equivalence is true for any and all such values. Therefore, we have
9766:
1628:
points, this means that any two codewords of the Reed–Solomon code disagree in at least
553:
use similar schemes. In the CD, two layers of Reed–Solomon coding separated by a 28-way
149:
22329:
22213:
22192:
22011:
21966:
21919:
21742:
21519:
19415:
12693:
are known, the error values can be determined. This can be done by direct solution for
10213:
9789:
9283:{\displaystyle (\alpha ^{j})^{i_{k}}=\alpha ^{j*i_{k}}=(\alpha ^{i_{k}})^{j}=X_{k}^{j}}
9088:
Then the syndromes can be written in terms of these error locators and error values as
7725:
7652:
7404:
7384:
7251:
7231:
7185:
7025:
6917:
6870:
6847:
6827:
6295:
6275:
5717:
5386:
5383:
to find the remainder, and then compensating for that remainder by subtracting it. The
5043:
3587:
3356:
3212:
3126:
3098:
3092:
2715:
2695:
2607:
2394:
2322:
2275:
2069:
1585:
1267:
1247:
1223:
1203:
1073:
1053:
1033:
975:
952:
929:
731:
634:
546:
412:
21613:
21596:
17205:. The Reed–Solomon code achieves this bound with equality, and can thus correct up to
12998:{\displaystyle g(x)=(x-3)(x-3^{2})(x-3^{3})(x-3^{4})=x^{4}+809x^{3}+723x^{2}+568x+522}
2066:, that is, the original message is always contained as a subsequence of the codeword.
1711:
114:
97:
22423:
22400:
22316:
22296:
22175:
21984:
21937:
21799:
21697:
21687:
21663:
21577:
21567:
21542:
836:
727:
472:
407:
266:
60:
3779:
is essentially the same as the encoding procedure; it uses the generator polynomial
3120:
617:
Reed–Solomon coding is less common in one-dimensional bar codes, but is used by the
521:
In 2002, another original scheme decoder was developed by Shuhong Gao, based on the
358:
There are two basic types of Reed–Solomon codes – original view and
22363:
22263:
22236:
22217:
22205:
22159:
22057:
21976:
21929:
21893:
21746:
21734:
21608:
21511:
14104:
12705:
6106:
4111:{\displaystyle a_{0},\dots ,a_{n-1}=\{1,\alpha ,\alpha ^{2},\dots ,\alpha ^{n-1}\}}
822:
764:
382:. Their seminal article was titled "Polynomial Codes over Certain Finite Fields". (
21632:
12574:, but that number has not been determined yet. The PGZ decoder does not determine
1095:
22390:
22333:
22308:
22288:
22251:
22227:(1960), "Encoding and Error Correction Procedures for the Bose-Chaudhuri Codes",
22224:
21980:
17323:
17167:
16700:{\displaystyle R_{j}=E_{j}=S_{j}=r(\alpha ^{j})\qquad {\text{for }}1\leq j\leq t}
14366:{\displaystyle \Delta =S_{i}+\Lambda _{1}\ S_{i-1}+\cdots +\Lambda _{e}\ S_{i-e}}
6246:
6162:
4952:
3581:
3152:
2063:
1893:
744:
737:
720:
467:
Reed–Solomon codes are used. Today, Reed–Solomon codes are widely implemented in
452:
423:
375:
238:
64:
45:
13470:{\displaystyle r(x)=s(x)+e(x)=3x^{6}+2x^{5}+123x^{4}+456x^{3}+191x^{2}+487x+474}
7535:
is also a polynomial of the same degree, this function gives rise to a codeword
5234:. To get a code that is overall systematic, we construct the message polynomial
22448:
Introduction to Reed–Solomon codes: principles, architecture and implementation
22413:
22247:
21958:
21467:
17237:
algorithm) and is based on interpolation and factorization of polynomials over
7694:
7646:
6323:
The theoretical error bound can be described via the following formula for the
2625:
371:
286:
234:
41:
22475:
21738:
15865:
The extended Euclidean algorithm can find a series of polynomials of the form
7674:
7018:
symbols per block. (This is a very popular value because of the prevalence of
661:
symbols of data, being generated, that are then sent over an erasure channel.
241:
in 1960. They have many applications, including consumer technologies such as
22557:
22367:
22240:
22209:
21701:
21546:
17319:
17280:
17234:
13145:{\displaystyle s_{r}(x)=p(x)\,x^{t}{\bmod {g}}(x)=547x^{3}+738x^{2}+442x+455}
8126:{\displaystyle s(x){\bmod {(}}x-\alpha ^{j})=g(x){\bmod {(}}x-\alpha ^{j})=0}
7019:
6867:
For practical uses of Reed–Solomon codes, it is common to use a finite field
4420:
that is known to both the sender and the receiver. The generator polynomial
4133:
In this view, the message is interpreted as the coefficients of a polynomial
756:
748:
637:-RS, can be used to overcome the unreliable nature of data transmission over
630:
22300:
22061:
21933:
21581:
13784:{\displaystyle S_{2}=r(3^{2})=637,\;S_{3}=r(3^{3})=762,\;S_{4}=r(3^{4})=925}
3955:
The inverse Fourier transform could be used to convert an error free set of
22188:
21829:
21432:
17453:% Multiply the information by X^(n-k), or just pad with zeros at the end to
15862:
The middle terms are zero due to the relationship between Λ and syndromes.
12675:
9724:
9166:
This definition of the syndrome values is equivalent to the previous since
4128:
3147:
Systematic encoding procedure: The message as an initial sequence of values
1028:
542:
460:
427:
329:
297:
278:
22454:
A Tutorial on Reed–Solomon Coding for Fault-Tolerance in RAID-like Systems
22077:"Open Source Soft-Decision Decoder for the JT65 (63,12) Reed–Solomon Code"
21911:
18611:% Prepare a linear system to solve the error polynomial and find the error
18551:% basically is nothing that we could do and we return the received message
17534:% Find the Reed-Solomon generating polynomial g(x), by the way this is the
8216:
The transmitted polynomial is corrupted in transit by an error polynomial
7188:, is a transparent code. This means that if the channel symbols have been
3806:
to map a set of evaluation points into the message values as shown above:
3319:{\displaystyle p_{m}(a_{i})=m_{i}{\text{ for all }}i\in \{0,\dots ,k-1\}.}
355:
is up to the designer of the code and may be selected within wide limits.
22492:
22349:
21963:
2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)
21651:
21437:
19181:(and optionally an error locating polynomial) from the received message.
18149:% Do the euclidean algorithm on the polynomials r0(x) and Syndromes(x) in
17303:
17221:
7670:
7200:
7189:
7178:
7072:
symbols in the block is a design parameter. A commonly used code encodes
6239:
6087:
3116:
1091:
893:
557:
554:
416:
337:
262:
22548:
22275:
1192:
Reed & Solomon's original view: The codeword as a sequence of values
668:
codewords received at the other end is enough to reconstruct all of the
21659:
21523:
19331:, and the number of subsets is infeasible for even modest codes. For a
17311:
14385:
has a detailed description of the procedure. In the following example,
906:
771:
611:
603:
491:
22163:
21897:
17966:% Prepare for the euclidean algorithm (Used to find the error locating
17870:% If all syndromes are zeros (perfectly divisible) there are no errors
16273:
Using the same data as the Peterson–Gorenstein–Zierler example above:
14405:
Using the same data as the Peterson Gorenstein Zierler example above:
8539:
The decoder starts by evaluating the polynomial as received at points
586:(discontinued in 2015) also used Reed–Solomon when breaking up files.
21720:"The development of turbo and LDPC codes for deep-space applications"
10039:. This follows from the above product notation construction since if
7174:
code, and is capable of correcting up to 16 symbol errors per block.
403:
22533:
22267:
21515:
18548:% If no error position is found we return the received work, because
17801:% Find the syndromes (Check if dividing the message by the generator
2070:
Simple encoding procedure: The message as a sequence of coefficients
2058:
by interpolating these values with a polynomial of degree less than
22016:
21971:
21924:
21916:
Proceedings of the 55th Annual ACM Symposium on Theory of Computing
21417:
17197:
was usually understood to limit the error-correction capability to
6166:
1684:
positions. Furthermore, there are two polynomials that do agree in
599:
575:
476:
395:
359:
242:
22539:
Matlab implementation of errors and-erasures Reed–Solomon decoding
22295:, International Symposium on Information Theory, San Remo, Italy,
22005:
17483:% Get the remainder of the division of the extended message by the
296:
Reed–Solomon codes operate on a block of data treated as a set of
22518:
22500:
12594:
5263:
by interpreting the message as the sequence of its coefficients.
5060:
largest monomials are equal to the corresponding coefficients of
770:
These concatenated codes are now being replaced by more powerful
760:
618:
607:
595:
578:
files which are commonly posted accompanying multimedia files on
258:
254:
22528:
21594:
19498:
Errors in transmission might cause this to be received instead.
16548:), and since a discrete Fourier transform is a linear operator,
13323:
Errors in transmission might cause this to be received instead.
2000:, whereas subsequent constructions interpret the message as the
711:
Deep-space concatenated coding system. Notation: RS(255, 223) +
444:
21910:
Brakensiek, Joshua; Gopi, Sivakanth; Makam, Visu (2023-06-02).
21595:
Sugiyama, Y.; Kasahara, M.; Hirasawa, S.; Namekawa, T. (1975).
17341:
17142:) and take the inverse transform (polynomial interpolation) of
12829:
7673:. Reed–Solomon codes in the BCH view are always cyclic because
707:
579:
499:
290:
21800:"Kissing Numbers, Sphere Packings, and Some Unexpected Proofs"
17306:'s confidence in the correctness of the symbol. The advent of
7098:
eight-bit data symbols plus 32 eight-bit parity symbols in an
5685:
are zero. Therefore, the following definition of the codeword
3377:. Once it has been found, it is evaluated at the other points
1953:
While the number of different polynomials of degree less than
560:
yields a scheme called Cross-Interleaved Reed–Solomon Coding (
347:
consecutive bit errors can affect at most two symbols of size
22256:
Journal of the Society for Industrial and Applied Mathematics
22004:
Alrabiah, Omar; Guruswami, Venkatesan; Li, Ray (2023-08-18),
21717:
18455:% Find the zeros of the error polynomial on this galois field
18314:% Check if the degree of remainder(x) is less than max_errors
17161:
13077:
8089:
8045:
6914:
elements. In this case, each symbol can be represented as an
5497:
3770:
1419:
of codewords of the Reed–Solomon code is defined as follows:
688:
583:
484:
274:
21597:"A method for solving key equation for decoding Goppa codes"
21157:
001 x + 908 x + 175 x + 194 x + 695 x + 094 x + 720 x + 000
17569:% A multiplication on the galois field is just a convolution
12587:
is reduced by one and the next smaller system is examined. (
22529:
Henry Minsky's RSCode library, Reed–Solomon encoder/decoder
22481:
21912:"Generic Reed-Solomon Codes Achieve List-Decoding Capacity"
17217:
errors. However, this error-correction bound is not exact.
16439:
12750:. This is generally done using a precomputed lookup table.
9311:
unknowns, but that system of equations is nonlinear in the
5524:{\displaystyle s_{r}(x)=p(x)\cdot x^{t}\ {\bmod {\ }}g(x).}
3151:
There is an alternative encoding procedure that produces a
684:
21965:. FOCS 2023, Santa Cruz, CA, USA, 2023. pp. 164–176.
20881:
Using the same data as the Berlekamp Welch example above:
17366:% RSENCODER Encode message with the Reed-Solomon algorithm
12832:
barcodes) for a RS(7,3) code. The generator polynomial is
8585:. We call the results of that evaluation the "syndromes",
6973:. Thus a Reed–Solomon code operating on 8-bit symbols has
6086:
The Reed–Solomon code is a code; in other words, it is a
5899:, that is, they are divisible by the generator polynomial
22523:
22315:(Revised ed.), Laguna Hills, CA: Aegean Park Press,
19440:
Using RS(7,3), GF(929), and the set of evaluation points
12570:
The above assumes the decoder knows the number of errors
8486:
The goal of the decoder is to find the number of errors (
8477:{\displaystyle e(x)=\sum _{k=1}^{\nu }e_{i_{k}}x^{i_{k}}}
6315:
550:
422:
In 1969, an improved BCH scheme decoder was developed by
270:
250:
17372:% prim_poly: Primitive polynomial p(x). Ie for DM is 301
14097:
R(x) = 001 x + 821 x + 329, with roots 27 = 3 and 81 = 3
7490:{\displaystyle (p(\alpha ^{1}),\dots ,p(\alpha ^{q-1}))}
4129:
The BCH view: The codeword as a sequence of coefficients
3209:
is defined as the unique polynomial of degree less than
22254:(1960), "Polynomial Codes over Certain Finite Fields",
21798:
Pfender, Florian; Ziegler, Günter M. (September 2004),
17510:% Now return the message with the redundant information
17233:. This algorithm produces a list of codewords (it is a
9318:
and does not have an obvious solution. However, if the
9079:{\displaystyle X_{k}=\alpha ^{i_{k}},\ Y_{k}=e_{i_{k}}}
7995:{\displaystyle g(x)=\prod _{j=1}^{n-k}(x-\alpha ^{j}),}
4875: is a polynomial that has at least the roots
3155:
Reed–Solomon code. Here, we use a different polynomial
2265:{\displaystyle p_{m}(a)=\sum _{i=0}^{k-1}m_{i}a^{i}\,.}
1957:
and the number of different messages are both equal to
402:
Also in 1960, a practical fixed polynomial decoder for
246:
21774:"Analytical Expressions Used in bercoding and BERTool"
20691:
20578:
20307:
20221:
20108:
19837:
19257:
19175:
18929:% Now to fix the errors just add with the encoded code
15695:
15090:
14231:{\displaystyle e_{1}x^{3}+e_{2}x^{4}=74x^{3}+122x^{4}}
14044:
14001:
13965:
13916:
13881:
13838:
13802:
12477:
12407:
12223:
9634:
9570:
9344:
9157:{\displaystyle S_{j}=\sum _{k=1}^{\nu }Y_{k}X_{k}^{j}}
7708:
7598:{\displaystyle (p(\alpha ^{2}),\dots ,p(\alpha ^{q}))}
6669:
6114:
3833:
3458:
3017:
2768:
2484:
629:
Specialized forms of Reed–Solomon codes, specifically
21098:
21061:
20997:
20967:
20890:
20301:
19831:
19607:
19525:
19418:
19382:
19337:
19225:
19198:
17243:
16838:
16811:
16784:
16742:
16715:
16617:
15084:
14993:
14632:
14276:
14152:
13959:
13796:
13658:
13491:
13329:
13157:
13030:
12838:
12729:
12645:
12618:
12217:
12062:
12029:
11889:
11565:
11545:
11221:
10990:
10316:
10268:
10236:
10216:
10164:
10086:
10045:
10010:
9841:
9792:
9769:
9740:
9338:
9172:
9094:
9009:
8950:
8921:
8879:
8600:
8545:
8403:
8296:
8238:
8139:
8031:
7924:
7827:
7806:{\displaystyle (c_{0},\ldots ,c_{i},\ldots ,c_{n-1})}
7742:
7655:
7611:
7541:
7503:
7427:
7407:
7387:
7351:
7301:
7274:
7254:
7234:
7210:
7130:
7104:
7078:
7048:
7028:
6979:
6940:
6920:
6893:
6873:
6850:
6830:
6782:
6724:
6517:
6337:
6298:
6278:
6204:
6178:
5934:
5905:
5883:
5854:
5769:
5740:
5720:
5691:
5649:
5565:
5539:
5445:
5409:
5403:
check symbols are created by computing the remainder
5389:
5360:
5328:
5301:
5272:
5240:
5211:
5182:
5153:
5124:
5095:
5066:
5046:
5017:
4961:
4742:
4714:
4474:
4455:
4426:
4400:
4371:
4342:
4313:
4307:
is constructed by multiplying the message polynomial
4284:
4255:
4223:
4197:
4168:
4139:
4016:
3977:
3812:
3785:
3746:
3694:
3610:
3590:
3437:
3383:
3359:
3332:
3235:
3215:
3188:
3161:
3129:
3101:
2738:
2718:
2698:
2672:
2634:
2610:
2463:
2417:
2397:
2345:
2325:
2298:
2278:
2188:
2161:
2084:
2014:
1963:
1906:
1864:
1851:{\displaystyle \delta =d/n=1-k/n+1/n=1-R+1/n\sim 1-R}
1758:
1720:
1690:
1634:
1608:
1588:
1425:
1403:
1302:
1270:
1250:
1226:
1206:
1162:
1136:
1103:
1076:
1056:
1036:
1001:
978:
955:
932:
21168:
055 x + 440 x + 497 x + 904 x + 424 x + 472 x + 001
16021:) don't need to be saved, so the algorithm becomes:
15061:{\displaystyle \Lambda (x)S(x)=Q(x)x^{t}+\Omega (x)}
14381:
so that a recalculated Δ would be zero. The article
13024:, then a systematic codeword is encoded as follows.
12701:
8207:{\displaystyle s(\alpha ^{j})=0,\ j=1,2,\ldots ,n-k}
7902:
As a result of the Reed-Solomon encoding procedure,
7697:
allows omitting some of the encoded parity symbols.
5011:, the encoder constructs the transmitted polynomial
3963:
message values back into the encoding polynomial of
1027:. The set of alphabet symbols is interpreted as the
475:
standards, though they are being slowly replaced by
451:
In 1977, Reed–Solomon codes were implemented in the
22172:
Error Control Coding: Fundamentals and Applications
22003:
21909:
21681:
17384:% enc_msg = rsEncoder(msg, 8, 301, 12, numel(msg));
16832:, the error locator polynomial, and these formulas
8378:will be zero if there is no error at that power of
399:scheme, and ones that use the BCH encoding scheme.
22549:Pure-Python implementation of a Reed–Solomon codec
22534:Open Source C++ Reed–Solomon Soft Decoding library
21117:
21083:
21038:
20983:
20952:
20749:
20279:
19816:
19585:
19424:
19404:
19355:
19323:
19238:
19211:
17268:
17108:
16824:
16797:
16755:
16728:
16699:
15852:
15060:
14977:
14365:
14230:
14067:
13939:
13783:
13645:
13469:
13315:
13144:
12997:
12735:
12666:
12631:
12562:
12183:
12048:
12013:
11873:
11551:
11529:
11205:
10966:
10302:
10254:
10222:
10200:
10150:
10072:
10031:
9986:
9798:
9778:
9752:
9698:
9282:
9156:
9078:
8978:
8963:
8936:
8907:
8865:
8577:
8476:
8361:
8283:
8206:
8125:
7994:
7892:
7805:
7661:
7637:
7597:
7527:
7489:
7413:
7393:
7373:
7337:
7287:
7260:
7240:
7216:
7166:
7116:
7090:
7060:
7034:
7010:
6965:
6926:
6906:
6879:
6856:
6844:is the symbol error rate in uncoded AWGN case and
6836:
6816:
6768:
6710:
6655:
6503:
6304:
6284:
6230:
6190:
6145:
6071:
5920:
5891:
5869:
5838:
5755:
5734:coefficients are identical to the coefficients of
5726:
5706:
5677:
5636:{\displaystyle x^{t-1},x^{t-2},\dots ,x^{1},x^{0}}
5635:
5551:
5523:
5431:
5395:
5375:
5346:
5314:
5287:
5266:Formally, the construction is done by multiplying
5255:
5226:
5197:
5168:
5139:
5110:
5081:
5052:
5032:
5003:
4936:
4726:
4698:
4461:
4441:
4412:
4386:
4354:
4328:
4299:
4270:
4241:
4209:
4183:
4154:
4110:
4003:
3945:
3798:
3759:
3732:
3681:{\displaystyle p_{m}(a_{0}),\dots ,p_{m}(a_{k-1})}
3680:
3596:
3570:
3421:
3365:
3345:
3318:
3221:
3201:
3174:
3135:
3107:
3081:
2724:
2704:
2684:
2658:
2616:
2596:
2449:
2403:
2383:
2331:
2311:
2284:
2264:
2174:
2147:
2046:
1976:
1938:
1884:
1850:
1744:
1702:
1676:
1620:
1594:
1570:
1411:
1334:
1276:
1256:
1232:
1212:
1180:
1148:
1122:
1082:
1062:
1042:
1019:
984:
961:
938:
479:. For example, Reed–Solomon codes are used in the
22388:
22380:, MIT Lecture Notes 6.451 (Video), archived from
21501:
17663:% RSDECODER Decode a Reed-Solomon encoded message
17420:% Get the Reed-Solomon generating polynomial g(x)
6597:
6584:
6445:
6432:
4811:
2457:for the Reed–Solomon code is defined as follows:
2148:{\displaystyle m=(m_{0},\dots ,m_{k-1})\in F^{k}}
1559:
1528:
1436:
505:In 1986, an original scheme decoder known as the
22555:
22477:Tutorial on Reed–Solomon Error Correction Coding
22143:Hong, Jonathan; Vetterli, Martin (August 1995),
20953:{\displaystyle R_{-1}=\prod _{i=1}^{n}(x-a_{i})}
19324:{\textstyle {\binom {n}{k}}={n! \over (n-k)!k!}}
18368:% Update r0, r1, g0, g1 and remove leading zeros
12807:Consider the Reed–Solomon code defined in
12711:
10158:, making the whole polynomial evaluate to zero.
8362:{\displaystyle e(x)=\sum _{i=0}^{n-1}e_{i}x^{i}}
8025:), it follows that it "inherits" all its roots.
7893:{\displaystyle s(x)=\sum _{i=0}^{n-1}c_{i}x^{i}}
7813:, is viewed as the coefficients of a polynomial
6694:
6120:
5839:{\displaystyle s(x)=p(x)\cdot x^{t}-s_{r}(x)\,.}
4946:
759:missions, where they perform within about 1–1.5
340:bit-error correcting codes, since a sequence of
22544:Octave implementation in communications package
21713:
21711:
17669:% = rsDecoder(enc_msg, 8, 301, 12, numel(msg))
10151:{\displaystyle (1-X_{k}^{-1}\cdot X_{k})=1-1=0}
8382:and nonzero if there is an error. If there are
3119:, and in the classical encoding procedure, its
21871:
21797:
21559:
12595:Find the roots of the error locator polynomial
10080:then one of the multiplied terms will be zero
8578:{\displaystyle \alpha ^{1}\dots \alpha ^{n-k}}
7248:, then by definition all non-zero elements of
574:Reed–Solomon error correction is also used in
541:Reed–Solomon coding is a key component of the
328:erroneous symbols at unknown locations. As an
22459:Algebraic soft-decoding of Reed–Solomon codes
22075:Franke, Steven J.; Taylor, Joseph H. (2016).
21634:New Algorithm For Decoding Reed-Solomon Codes
19274:
19261:
18860:% Put the error magnitude on the error vector
18152:% order to find the error locating polynomial
12681:
12678:is an efficient implementation of this step.
3115:. In other words, the Reed–Solomon code is a
1520:
1444:
778:Channel coding schemes used by NASA missions
594:Almost all two-dimensional bar codes such as
370:Reed–Solomon codes were developed in 1960 by
207:
22441:
22142:
22074:
22035:
21807:Notices of the American Mathematical Society
21708:
21179:702 x + 845 x + 691 x + 461 x + 327 x + 237
21033:
20998:
17456:% get space to add the redundant information
8497:), and the error values at those positions (
7332:
7308:
4105:
4055:
3310:
3286:
693:Space Communications Protocol Specifications
641:. The encoding process assumes a code of RS(
22348:
22246:
22193:"Shift-register synthesis and BCH decoding"
22170:Lin, Shu; Costello, Jr., Daniel J. (1983),
22169:
21760:
21560:Peterson, W. Wesley; Weldon, E.J. (1996) .
21490:
19189:
14253:
13477:The syndromes are calculated by evaluating
12197:was chosen to be any integer between 1 and
9809:
7910:) is divisible by the generator polynomial
4162:. The sender computes a related polynomial
2075:
1985:
1197:
917:
715:("constraint length" = 7, code rate = 1/2).
383:
22524:Schifra Open Source C++ Reed–Solomon Codec
19366:
17537:% same as the rsgenpoly function on matlab
17162:Decoding beyond the error-correction bound
13742:
13700:
6711:{\textstyle t={\frac {1}{2}}(d_{\min }-1)}
4816:
4808:
3771:Discrete Fourier transform and its inverse
1556:
1533:
1525:
1441:
281:systems such as satellite communications,
214:
200:
22464:Wikiversity:Reed–Solomon codes for coders
22352:(October 1965), "On Decoding BCH Codes",
22307:
22287:
22051:
22015:
21970:
21923:
21887:
21684:Reed Solomon Codes and Their Applications
21656:Reed–Solomon Codes and Their Applications
21612:
17486:% Reed-Solomon generating polynomial g(x)
16261:(0) is the constant (low order) term of A
14103:To calculate the error values, apply the
13185:
13065:
9727:), this is the end. The error locations (
9708:Consequently, the problem is finding the
6817:{\displaystyle h={\frac {m}{\log _{2}M}}}
6065:
6052:
6051:
5832:
5205:are a subsequence of the coefficients of
2258:
1564:
582:. The distributed online storage service
22412:
22223:
21956:
21536:
19586:{\displaystyle b_{i}E(a_{i})-Q(a_{i})=0}
18998:% Remove leading zeros from Galois array
16500:) as the discrete Fourier transforms of
16440:Decoder using discrete Fourier transform
8515:) can be calculated and subtracted from
7194:see 'Remarks' at the end of this section
6314:
6312:is the number of erasures in the block.
5354:check symbols, dividing that product by
1710:points but are not equal, and thus, the
706:
443:
22373:
22355:IEEE Transactions on Information Theory
22277:The Original View of Reed–Solomon Codes
22201:IEEE Transactions on Information Theory
22039:IEEE Transactions on Information Theory
21875:IEEE Transactions on Information Theory
18896:% Bring this vector to the galois field
14077:Λ(x) = 329 x + 821 x + 001, with roots
7638:{\displaystyle \alpha ^{q}=\alpha ^{1}}
2411:. Thus the classical encoding function
726:Voyager introduced Reed–Solomon coding
515:Guruswami–Sudan list decoding algorithm
14:
22556:
22473:
22419:Error-Control Coding for Data Networks
22229:IRE Transactions on Information Theory
22187:
21650:
21407:= {001, 006, 017, 034, 057, 086, 121}
20864:= {001, 006, 017, 034, 057, 086, 121}
19513:= {001, 006, 123, 456, 057, 086, 121}
19494:= {001, 006, 017, 034, 057, 086, 121}
19184:
17381:% Example: msg = uint8('Test')
11559:that are unaffected by the summation.
10201:{\displaystyle \Lambda (X_{k}^{-1})=0}
7649:of the original codeword derived from
5118:are chosen exactly in such a way that
5089:, and the lower-order coefficients of
477:Bose–Chaudhuri–Hocquenghem (BCH) codes
22328:
22273:
22124:, Stanford University, archived from
21626:
21624:
21400:resulting in the corrected codeword:
21277:) by most significant coefficient of
20857:resulting in the corrected codeword:
12791:) to get the originally sent message
8523:) to get the originally sent message
8224:) to produce the received polynomial
7715:Peterson–Gorenstein–Zierler algorithm
7338:{\displaystyle i\in \{1,\dots ,q-1\}}
6167:maximum distance separable (MDS) code
702:
22498:
22396:The Theory of Error-Correcting Codes
22145:"Simple Algorithms for BCH Decoding"
22114:
21190:266 x + 086 x + 798 x + 311 x + 532
17369:% m is the number of bits per symbol
14590:
14110:Ω(x) = S(x) Λ(x) mod x = 546 x + 732
12588:
11215:Collect each term into its own sum.
8971:, as shown in the previous section.
8534:
7700:
7528:{\displaystyle a\mapsto p(\alpha a)}
7124:-symbol block; this is denoted as a
3733:{\displaystyle m_{0},\dots ,m_{k-1}}
3422:{\displaystyle a_{k},\dots ,a_{n-1}}
2384:{\displaystyle a_{0},\dots ,a_{n-1}}
1714:of the Reed–Solomon code is exactly
683:Reed–Solomon codes are also used in
624:
22334:"The Ubiquitous Reed–Solomon Codes"
22152:IEEE Transactions on Communications
21630:
19176:Reed Solomon original view decoders
17378:% n is the total size (k+redundant)
14583:is the error locator polynomial, Λ(
14266:) with an assumed number of errors
14246:) reproduces the original codeword
10303:{\displaystyle Y_{k}X_{k}^{j+\nu }}
7709:Peterson–Gorenstein–Zierler decoder
1939:{\displaystyle \delta +R\leq 1+1/n}
457:concatenated error correction codes
432:Berlekamp–Massey decoding algorithm
24:
22512:
22474:Geisel, William A. (August 1990),
22108:
21621:
21039:{\displaystyle \{a_{i},b(a_{i})\}}
19265:
17052:
17017:
16988:
16932:
16897:
16869:
15833:
15816:
15792:
15641:
15592:
15569:
15520:
15497:
15474:
15425:
15402:
15379:
15330:
15307:
15284:
15235:
15212:
15189:
15153:
15130:
15094:
15046:
14994:
14962:
14946:
14905:
14882:
14862:
14840:
14799:
14776:
14756:
14335:
14297:
14277:
14019:
14005:
13856:
13842:
12452:
12425:
12411:
12211:of the error location polynomial:
12150:
12103:
12074:
11986:
11951:
11910:
11800:
11713:
11632:
11546:
11482:
11395:
11314:
11158:
11102:
11052:
10920:
10864:
10814:
10710:
10642:
10580:
10502:
10465:
10434:
10353:
10165:
9965:
9936:
9913:
9842:
6588:
6511:and for other modulation schemes:
6436:
5040:such that the coefficients of the
3182:. In this variant, the polynomial
2047:{\displaystyle a_{1},\dots ,a_{k}}
1335:{\displaystyle a_{1},\dots ,a_{n}}
1284:elements. In turn, the polynomial
430:, and has since been known as the
390:(unencoded message length) out of
25:
22580:
22436:
19595:Assume maximum number of errors:
17330:
17170:states that the minimum distance
16611:) are the same as the syndromes:
12753:
12704:matrix given above, or using the
8017:) is a multiple of the generator
7199:Whether the Reed–Solomon code is
6769:{\displaystyle P_{s}=1-(1-s)^{h}}
5533:The remainder has degree at most
4004:{\displaystyle a_{i}=\alpha ^{i}}
885:Cassini, Mars Pathfinder, others
378:, who were then staff members of
21957:Guo, Zeyu; Zhang, Zihan (2023).
21423:Berlekamp–Massey algorithm
19371:In 1986, a decoder known as the
17804:% polynomial the result is zero)
17296:
17174:of a linear block code of size (
16476:) are the same as above. Define
14383:Berlekamp–Massey algorithm
14260:Berlekamp–Massey algorithm
10255:{\displaystyle 1\leq j\leq \nu }
8908:{\displaystyle s(\alpha ^{j})=0}
8490:), the positions of the errors (
7184:The Reed–Solomon code, like the
6249:. A Reed–Solomon code (like any
5885:
5714:has the property that the first
4744:
4122:
2450:{\displaystyle C:F^{k}\to F^{n}}
2074:In the original construction of
1948:maximum distance separable codes
1752:. Then the relative distance is
1582:polynomials of degree less than
1539: is a polynomial over
1427:
1405:
1123:{\displaystyle R={\frac {k}{n}}}
22093:from the original on 2017-03-09
22068:
22029:
21997:
21950:
21903:
21865:
21856:
21844:
21835:
21820:from the original on 2008-05-09
21791:
21780:from the original on 2019-02-01
21766:
21753:
21460:
19599:= 2. The key equations become:
17344:implementation for an encoder.
16778:) using the known coefficients
16676:
11539:Extract the constant values of
9293:The syndromes give a system of
8979:Error locators and error values
8825:
7167:{\displaystyle (n,k)=(255,223)}
6165:. Such a code is also called a
6047:
5678:{\displaystyle p(x)\cdot x^{t}}
2004:of the polynomial at the first
898:Messenger, Stereo, MRO, others
800:Explorer, Mariner, many others
533:
528:
190:Maximum-distance separable code
22564:Error detection and correction
22469:BBC R&D White Paper WHP031
21675:
21644:
21588:
21553:
21530:
21495:
21484:
21030:
21017:
20947:
20928:
20868:
19780:
19660:
19654:
19618:
19574:
19561:
19552:
19539:
19399:
19386:
19350:
19338:
19306:
19294:
17375:% k is the size of the message
17263:
17250:
17077:
16984:
16957:
16880:
16673:
16660:
15929:increases. Once the degree of
15055:
15049:
15030:
15024:
15015:
15009:
15003:
14997:
14871:
14865:
14765:
14759:
14646:
14640:
14254:Berlekamp–Massey decoder
13772:
13759:
13730:
13717:
13688:
13675:
13521:
13508:
13369:
13363:
13354:
13348:
13339:
13333:
13215:
13209:
13182:
13176:
13167:
13161:
13092:
13086:
13062:
13056:
13047:
13041:
12932:
12913:
12910:
12891:
12888:
12869:
12866:
12854:
12848:
12842:
10377:
10356:
10189:
10168:
10127:
10087:
9900:
9878:
9851:
9845:
9253:
9232:
9187:
9173:
8975:was not corrupted in transit.
8931:
8925:
8896:
8883:
8739:
8726:
8710:
8697:
8682:
8669:
8660:
8647:
8638:
8625:
8413:
8407:
8306:
8300:
8284:{\displaystyle r(x)=s(x)+e(x)}
8278:
8272:
8263:
8257:
8248:
8242:
8156:
8143:
8114:
8093:
8085:
8079:
8070:
8049:
8041:
8035:
7986:
7967:
7934:
7928:
7837:
7831:
7800:
7743:
7731:
7592:
7589:
7576:
7561:
7548:
7542:
7522:
7513:
7507:
7484:
7481:
7462:
7447:
7434:
7428:
7367:
7359:
7161:
7149:
7143:
7131:
7022:computer systems.) The number
6757:
6744:
6705:
6686:
6638:
6618:
6486:
6466:
6217:
6205:
6062:
6056:
6038:
6032:
6016:
6010:
5994:
5988:
5959:
5953:
5944:
5938:
5915:
5909:
5864:
5858:
5829:
5823:
5794:
5788:
5779:
5773:
5750:
5744:
5701:
5695:
5659:
5653:
5559:, whereas the coefficients of
5515:
5509:
5477:
5471:
5462:
5456:
5426:
5420:
5370:
5364:
5282:
5276:
5250:
5244:
5221:
5215:
5192:
5186:
5163:
5157:
5134:
5128:
5105:
5099:
5076:
5070:
5027:
5021:
4998:
4992:
4986:
4980:
4971:
4965:
4826:
4820:
4484:
4478:
4436:
4430:
4381:
4375:
4323:
4317:
4294:
4288:
4265:
4259:
4178:
4172:
4149:
4143:
3932:
3913:
3889:
3876:
3859:
3846:
3822:
3816:
3675:
3656:
3634:
3621:
3557:
3538:
3514:
3501:
3484:
3471:
3447:
3441:
3259:
3246:
2748:
2742:
2644:
2638:
2583:
2564:
2540:
2527:
2510:
2497:
2473:
2467:
2434:
2205:
2199:
2129:
2091:
1653:
1641:
1515:
1502:
1487:
1474:
1465:
1452:
852:Voyager, Galileo, many others
494:, but BCH codes are used with
289:, and storage systems such as
13:
1:
21631:Gao, Shuhong (January 2002),
21614:10.1016/S0019-9958(75)90090-X
21477:
19464:If the message polynomial is
17723:% Initialize the error vector
16770:= number of errors. Generate
14238:from the received polynomial
13005:If the message polynomial is
12712:Calculate the error locations
7011:{\displaystyle n=2^{8}-1=255}
6146:{\textstyle d_{\min }=n-k+1.}
6081:
5004:{\displaystyle s(x)=p(x)g(x)}
4953:systematic encoding procedure
4947:Systematic encoding procedure
1677:{\displaystyle n-(k-1)=n-k+1}
22499:Reid, Jeff A. (April 1995),
21981:10.1109/FOCS57990.2023.00019
21763:, p. 171), for example.
21537:Peterson, W. Wesley (1961).
20876:
19435:
18293:% Update g as g0-quotient*g1
16268:
14597:extended Euclidean algorithm
14400:
12783:and then is subtracted from
10073:{\displaystyle x=X_{k}^{-1}}
8983:For convenience, define the
7645:holds, this codeword is the
6292:is the number of errors and
5892:{\displaystyle \mathbf {C} }
2155:is mapped to the polynomial
1412:{\displaystyle \mathbf {C} }
1020:{\displaystyle k<n\leq q}
523:extended Euclidean algorithm
439:extended Euclidean algorithm
7:
22118:EE387 Notes #7, Handout #28
21686:. IEEE Press. p. 433.
21566:(2nd ed.). MIT Press.
21411:
17658:encoded, m, prim_poly, n, k
16179:to set low order term of Λ(
10984:and it will still be zero.
10310:and it will still be zero.
8964:{\displaystyle \alpha ^{j}}
7288:{\displaystyle \alpha ^{i}}
5848:As a result, the codewords
5176:. Then the coefficients of
4708:For a "narrow sense code",
3326:To compute this polynomial
763:of the ultimate limit, the
589:
10:
22585:
21853:by W Wesley Peterson, 1961
20991:Lagrange interpolation of
18218:% Find quotient*g1 and pad
17640:
17335:
12802:
12799:), with errors corrected.
12682:Calculate the error values
12667:{\displaystyle X_{k}^{-1}}
12049:{\displaystyle S_{j+\nu }}
10032:{\displaystyle X_{k}^{-1}}
8386:errors at distinct powers
7712:
7680:
3777:discrete Fourier transform
2292:is obtained by evaluating
2054:and obtain the polynomial
871:RS(255, 223) + CC(14, 1/4)
481:Digital Video Broadcasting
365:
22442:Information and tutorials
21739:10.1109/JPROC.2007.905132
21491:Reed & Solomon (1960)
21428:Berlekamp–Welch algorithm
19373:Berlekamp–Welch algorithm
19356:{\displaystyle (255,249)}
19190:Reed & Solomon (1960)
18791:% Solve the linear system
17340:Here we present a simple
17269:{\displaystyle GF(2^{m})}
17231:algebraic geometric codes
14393:) is used to represent Λ(
10262:. Multiply both sides by
10230:be any integer such that
9760:errors can be corrected.
7736:The transmitted message,
7421:gives rise to a codeword
6966:{\displaystyle n=2^{m}-1}
6864:is the modulation order.
4249:and sends the polynomial
4242:{\displaystyle n\leq q-1}
2685:{\displaystyle n\times k}
2076:Reed & Solomon (1960)
1986:Reed & Solomon (1960)
1198:Reed & Solomon (1960)
860:RS(255, 223) + CC(7, 1/3)
849:RS(255, 223) + CC(7, 1/2)
507:Berlekamp–Welch algorithm
320:locate and correct up to
195:
188:
183:
168:
163:
148:
113:
96:
84:
72:
56:
51:
37:
32:
22502:CRC and Reed Solomon ECC
22480:, Technical Memorandum,
22416:; Chen, Xuemin (2012) ,
22368:10.1109/TIT.1965.1053825
22241:10.1109/TIT.1960.1057586
22210:10.1109/tit.1969.1054260
21761:Lin & Costello (1983
21453:
21448:Forward error correction
21443:Folded Reed–Solomon code
21084:{\displaystyle A_{-1}=0}
19460:= {0, 1, 2, 3, 4, 5, 6}
19405:{\displaystyle O(n^{3})}
17647:
17346:
17314:, which employ iterated
16460:) the encoded codeword.
12686:Once the error locators
11552:{\displaystyle \Lambda }
9826:error locator polynomial
9810:Error locator polynomial
8006:is a primitive element.
6161:); this is known as the
5432:{\displaystyle s_{r}(x)}
2628:, that is, it satisfies
1244:, over the finite field
918:Constructions (encoding)
697:forward error correction
487:, in conjunction with a
233:that were introduced by
22313:Algebraic Coding Theory
22062:10.1109/TIT.2003.819332
21934:10.1145/3564246.3585128
21727:Proceedings of the IEEE
21601:Information and Control
21118:{\displaystyle A_{0}=1}
20849:E(x) = 0 : {2, 3}
19517:The key equations are:
19367:Berlekamp Welch decoder
17645:Now the decoding part:
17361:msg, m, prim_poly, n, k
12736:{\displaystyle \alpha }
12723:by taking the log base
12639:(not their reciprocals
12056:from both sides yields
8594:. They are defined as:
7217:{\displaystyle \alpha }
6231:{\displaystyle (n-k)/2}
5877:are indeed elements of
4462:{\displaystyle \alpha }
2659:{\displaystyle C(m)=Am}
1988:interprets the message
1745:{\displaystyle d=n-k+1}
384:Reed & Solomon 1960
332:, it can correct up to
22374:Koetter, Ralf (2005),
22293:Nonbinary BCH decoding
22204:, IT-15 (1): 122–127,
21851:Error Correcting Codes
21563:Error Correcting Codes
21539:Error Correcting Codes
21119:
21085:
21040:
20985:
20984:{\displaystyle R_{0}=}
20954:
20927:
20751:
20281:
19818:
19587:
19426:
19406:
19357:
19325:
19240:
19213:
18434:% Remove leading zeros
17270:
17182:) is upper-bounded by
17110:
16826:
16799:
16757:
16730:
16701:
15854:
15062:
14979:
14367:
14232:
14069:
13941:
13785:
13647:
13471:
13317:
13146:
12999:
12737:
12668:
12633:
12599:Use the coefficients Λ
12564:
12185:
12050:
12015:
11875:
11834:
11747:
11666:
11591:
11553:
11531:
11480:
11393:
11312:
11247:
11207:
11011:
10968:
10304:
10256:
10224:
10202:
10152:
10074:
10033:
9988:
9877:
9800:
9786:, instead of just the
9780:
9754:
9700:
9284:
9158:
9128:
9080:
8965:
8938:
8909:
8867:
8772:
8579:
8478:
8439:
8363:
8338:
8285:
8208:
8127:
7996:
7966:
7894:
7869:
7807:
7722:Error Correcting Codes
7663:
7639:
7599:
7529:
7491:
7415:
7395:
7375:
7339:
7289:
7262:
7242:
7218:
7168:
7118:
7092:
7062:
7061:{\displaystyle k<n}
7036:
7012:
6967:
6928:
6908:
6881:
6858:
6838:
6818:
6770:
6712:
6657:
6577:
6505:
6425:
6320:
6306:
6286:
6243:signal-to-noise ratios
6232:
6192:
6147:
6073:
5922:
5893:
5871:
5840:
5757:
5728:
5708:
5679:
5637:
5553:
5525:
5433:
5397:
5377:
5348:
5316:
5289:
5257:
5228:
5199:
5170:
5141:
5112:
5083:
5054:
5034:
5005:
4938:
4852:
4728:
4700:
4463:
4443:
4414:
4388:
4356:
4330:
4301:
4272:
4243:
4211:
4185:
4156:
4112:
4005:
3947:
3800:
3761:
3734:
3682:
3598:
3572:
3423:
3375:Lagrange interpolation
3367:
3347:
3320:
3223:
3203:
3176:
3137:
3109:
3083:
2726:
2706:
2686:
2660:
2618:
2598:
2451:
2405:
2385:
2333:
2313:
2286:
2266:
2237:
2176:
2149:
2048:
1978:
1940:
1886:
1852:
1746:
1704:
1678:
1622:
1596:
1572:
1413:
1336:
1278:
1258:
1234:
1214:
1182:
1150:
1124:
1084:
1064:
1044:
1021:
986:
963:
940:
753:Mars Exploration Rover
716:
448:
380:MIT Lincoln Laboratory
231:error-correcting codes
27:Error-correcting codes
22274:Welch, L. R. (1997),
21120:
21086:
21041:
20986:
20955:
20907:
20752:
20282:
19819:
19588:
19427:
19407:
19358:
19326:
19241:
19239:{\displaystyle a_{n}}
19214:
19212:{\displaystyle a_{1}}
17271:
17111:
16827:
16825:{\displaystyle E_{t}}
16800:
16798:{\displaystyle E_{1}}
16758:
16756:{\displaystyle R_{t}}
16731:
16729:{\displaystyle R_{1}}
16702:
15855:
15063:
14987:The key equation is:
14980:
14368:
14233:
14070:
13942:
13786:
13648:
13472:
13318:
13147:
13000:
12738:
12669:
12634:
12632:{\displaystyle X_{k}}
12565:
12186:
12051:
12016:
11876:
11814:
11727:
11646:
11571:
11554:
11532:
11460:
11373:
11292:
11227:
11208:
10991:
10969:
10305:
10257:
10225:
10203:
10153:
10075:
10034:
9989:
9857:
9801:
9781:
9755:
9701:
9285:
9159:
9108:
9081:
8966:
8939:
8910:
8868:
8752:
8580:
8479:
8419:
8364:
8312:
8286:
8209:
8128:
7997:
7940:
7895:
7843:
7808:
7664:
7640:
7600:
7530:
7497:. Since the function
7492:
7416:
7396:
7376:
7374:{\displaystyle q=|F|}
7340:
7290:
7263:
7243:
7219:
7169:
7119:
7117:{\displaystyle n=255}
7093:
7091:{\displaystyle k=223}
7063:
7037:
7013:
6968:
6929:
6909:
6907:{\displaystyle 2^{m}}
6882:
6859:
6839:
6819:
6771:
6713:
6658:
6551:
6506:
6399:
6318:
6307:
6287:
6233:
6193:
6148:
6074:
5923:
5894:
5872:
5841:
5758:
5729:
5709:
5680:
5638:
5554:
5526:
5434:
5398:
5378:
5349:
5347:{\displaystyle t=n-k}
5322:to make room for the
5317:
5315:{\displaystyle x^{t}}
5290:
5258:
5229:
5200:
5171:
5147:becomes divisible by
5142:
5113:
5084:
5055:
5035:
5006:
4939:
4832:
4729:
4701:
4464:
4444:
4415:
4389:
4357:
4331:
4302:
4273:
4244:
4212:
4186:
4157:
4113:
4006:
3948:
3801:
3799:{\displaystyle p_{m}}
3762:
3760:{\displaystyle p_{m}}
3740:by the definition of
3735:
3683:
3599:
3573:
3424:
3368:
3348:
3346:{\displaystyle p_{m}}
3321:
3224:
3204:
3202:{\displaystyle p_{m}}
3177:
3175:{\displaystyle p_{m}}
3138:
3110:
3084:
2727:
2707:
2687:
2661:
2619:
2599:
2452:
2406:
2386:
2334:
2314:
2312:{\displaystyle p_{m}}
2287:
2267:
2211:
2177:
2175:{\displaystyle p_{m}}
2150:
2049:
1979:
1977:{\displaystyle q^{k}}
1941:
1887:
1885:{\displaystyle R=k/n}
1853:
1747:
1705:
1679:
1623:
1597:
1573:
1547: of degree
1414:
1337:
1279:
1259:
1235:
1215:
1183:
1181:{\displaystyle n=q-1}
1151:
1125:
1085:
1065:
1045:
1022:
987:
964:
941:
710:
657:symbols each storing
473:digital communication
447:
269:technologies such as
22389:MacWilliams, F. J.;
21392:) = 0 : {2, 3}
21285:) = 708. (Optional)
21193:708 x + 176 x + 532
21096:
21059:
20995:
20965:
20888:
20299:
20291:Gaussian elimination
19829:
19605:
19523:
19416:
19380:
19335:
19255:
19249:binomial coefficient
19223:
19196:
18161:% Do a long division
17276:and its extensions.
17241:
17226:Venkatesan Guruswami
16836:
16809:
16782:
16740:
16713:
16615:
15921:where the degree of
15082:
14991:
14630:
14274:
14150:
13957:
13951:Gaussian elimination
13794:
13656:
13489:
13327:
13155:
13028:
12836:
12766:) is generated from
12727:
12643:
12616:
12215:
12060:
12027:
11887:
11563:
11543:
11219:
10988:
10314:
10266:
10234:
10214:
10162:
10084:
10043:
10008:
10004:are the reciprocals
9839:
9790:
9767:
9738:
9336:
9170:
9092:
9007:
8948:
8937:{\displaystyle s(x)}
8919:
8877:
8598:
8543:
8401:
8294:
8236:
8137:
8029:
7922:
7825:
7740:
7675:BCH codes are cyclic
7653:
7609:
7539:
7501:
7425:
7405:
7385:
7349:
7299:
7272:
7252:
7232:
7208:
7128:
7102:
7076:
7046:
7026:
6977:
6938:
6918:
6891:
6871:
6848:
6828:
6780:
6722:
6667:
6515:
6335:
6296:
6276:
6272:is satisfied, where
6202:
6176:
6112:
5932:
5921:{\displaystyle g(x)}
5903:
5881:
5870:{\displaystyle s(x)}
5852:
5767:
5756:{\displaystyle p(x)}
5738:
5718:
5707:{\displaystyle s(x)}
5689:
5647:
5563:
5537:
5443:
5407:
5387:
5376:{\displaystyle g(x)}
5358:
5326:
5299:
5288:{\displaystyle p(x)}
5270:
5256:{\displaystyle p(x)}
5238:
5227:{\displaystyle s(x)}
5209:
5198:{\displaystyle p(x)}
5180:
5169:{\displaystyle g(x)}
5151:
5140:{\displaystyle s(x)}
5122:
5111:{\displaystyle s(x)}
5093:
5082:{\displaystyle p(x)}
5064:
5044:
5033:{\displaystyle s(x)}
5015:
4959:
4740:
4712:
4472:
4453:
4442:{\displaystyle g(x)}
4424:
4398:
4387:{\displaystyle g(x)}
4369:
4364:generator polynomial
4340:
4329:{\displaystyle p(x)}
4311:
4300:{\displaystyle s(x)}
4282:
4271:{\displaystyle s(x)}
4253:
4221:
4195:
4184:{\displaystyle s(x)}
4166:
4155:{\displaystyle p(x)}
4137:
4014:
3975:
3810:
3783:
3744:
3692:
3608:
3588:
3435:
3381:
3357:
3330:
3233:
3213:
3186:
3159:
3127:
3099:
2736:
2716:
2696:
2670:
2632:
2608:
2461:
2415:
2395:
2343:
2323:
2296:
2276:
2186:
2159:
2082:
2012:
1961:
1904:
1862:
1756:
1718:
1688:
1632:
1606:
1586:
1423:
1401:
1300:
1268:
1248:
1240:of degree less than
1224:
1204:
1160:
1134:
1101:
1074:
1054:
1034:
999:
976:
953:
930:
653:codewords of length
22495:(scholarpedia.org).
22330:Cipra, Barry Arthur
22309:Berlekamp, Elwyn R.
22289:Berlekamp, Elwyn R.
22225:Peterson, Wesley W.
22115:Gill, John (n.d.),
19813:
19779:
19751:
19723:
19185:Theoretical decoder
19108:% Add leading zeros
14579:The final value of
14373:and then adjusts Λ(
14113:Λ'(x) = 658 x + 821
12663:
11859:
11784:
11703:
11622:
11515:
11440:
11359:
11278:
11191:
11147:
11097:
11047:
10953:
10909:
10859:
10809:
10767:
10749:
10699:
10681:
10637:
10619:
10575:
10528:
10491:
10460:
10421:
10376:
10352:
10299:
10188:
10113:
10069:
10028:
9753:{\displaystyle n-k}
9556:
9528:
9505:
9458:
9436:
9419:
9400:
9378:
9361:
9279:
9153:
6617:
6465:
6245:)—these are called
6191:{\displaystyle n-k}
5552:{\displaystyle t-1}
4727:{\displaystyle i=1}
4413:{\displaystyle n-k}
4355:{\displaystyle k-1}
4336:, which has degree
4210:{\displaystyle n-1}
3277: for all
3003:
2969:
2894:
2866:
2830:
2802:
2712:with elements from
1703:{\displaystyle k-1}
1621:{\displaystyle k-1}
1149:{\displaystyle n=q}
911:Constellation, MSL
809:convolutional codes
779:
732:convolutional codes
664:Any combination of
649:) which results in
316:erroneous symbols,
22489:Concatenated codes
22377:Reed–Solomon Codes
22086:(May/June): 8–17.
21115:
21081:
21036:
20981:
20950:
20747:
20741:
20677:
20567:
20277:
20271:
20207:
20097:
19814:
19799:
19765:
19737:
19709:
19583:
19422:
19402:
19353:
19321:
19236:
19209:
17266:
17106:
17104:
16822:
16795:
16753:
16726:
16697:
15850:
15844:
15681:
15058:
14975:
14973:
14363:
14228:
14065:
14059:
14030:
13990:
13937:
13931:
13902:
13867:
13827:
13781:
13643:
13467:
13313:
13142:
12995:
12733:
12664:
12646:
12629:
12560:
12554:
12463:
12396:
12181:
12046:
12011:
11871:
11845:
11758:
11677:
11602:
11549:
11527:
11501:
11414:
11333:
11258:
11203:
11177:
11121:
11071:
11027:
10964:
10962:
10939:
10883:
10833:
10789:
10750:
10729:
10682:
10661:
10620:
10599:
10555:
10511:
10474:
10443:
10401:
10359:
10332:
10300:
10279:
10252:
10220:
10198:
10171:
10148:
10096:
10070:
10052:
10029:
10011:
9984:
9796:
9779:{\displaystyle 2v}
9776:
9750:
9696:
9690:
9620:
9559:
9536:
9508:
9485:
9444:
9422:
9405:
9386:
9364:
9347:
9280:
9265:
9154:
9139:
9076:
8961:
8934:
8905:
8863:
8861:
8575:
8474:
8359:
8281:
8204:
8123:
7992:
7890:
7803:
7726:W. Wesley Peterson
7659:
7635:
7595:
7525:
7487:
7411:
7391:
7381:. Each polynomial
7371:
7335:
7285:
7258:
7238:
7214:
7186:convolutional code
7164:
7114:
7088:
7058:
7032:
7008:
6963:
6924:
6904:
6877:
6854:
6834:
6814:
6766:
6708:
6653:
6603:
6501:
6451:
6321:
6302:
6282:
6228:
6188:
6143:
6069:
5918:
5889:
5867:
5836:
5753:
5724:
5704:
5675:
5643:in the polynomial
5633:
5549:
5521:
5429:
5393:
5373:
5344:
5312:
5285:
5253:
5224:
5195:
5166:
5137:
5108:
5079:
5050:
5030:
5001:
4934:
4724:
4696:
4459:
4439:
4410:
4384:
4352:
4326:
4297:
4268:
4239:
4207:
4181:
4152:
4108:
4001:
3943:
3937:
3796:
3757:
3730:
3678:
3594:
3568:
3562:
3419:
3363:
3343:
3316:
3219:
3199:
3172:
3133:
3105:
3093:Vandermonde matrix
3079:
3073:
3006:
2977:
2949:
2874:
2852:
2810:
2788:
2722:
2702:
2682:
2666:for the following
2656:
2614:
2594:
2588:
2447:
2401:
2381:
2329:
2309:
2282:
2262:
2172:
2145:
2044:
1996:of the polynomial
1974:
1936:
1882:
1848:
1742:
1700:
1674:
1618:
1592:
1568:
1409:
1397:Formally, the set
1332:
1274:
1254:
1230:
1210:
1178:
1146:
1120:
1080:
1060:
1040:
1017:
982:
959:
936:
777:
717:
703:Space transmission
498:in its successor,
449:
413:W. Wesley Peterson
227:Reed–Solomon codes
33:Reed–Solomon codes
22406:978-0-444-85010-2
22399:, North-Holland,
22322:978-0-89412-063-3
22181:978-0-13-283796-5
22174:, Prentice-Hall,
22164:10.1109/26.403765
22046:(11): 2809–2825.
21990:979-8-3503-1894-4
21943:978-1-4503-9913-5
21898:10.1109/18.782097
21669:978-0-7803-1025-4
21573:978-0-585-30709-1
21197:
21196:
19425:{\displaystyle n}
19319:
19272:
17320:theoretical limit
17085:
16878:
16680:
16397:
16396:
16183:) to 1, divide Λ(
14591:Euclidean decoder
14577:
14576:
14346:
14308:
12828:(this is used in
10223:{\displaystyle j}
9799:{\displaystyle v}
9045:
8535:Syndrome decoding
8170:
7701:BCH view decoders
7662:{\displaystyle p}
7647:cyclic left-shift
7414:{\displaystyle F}
7394:{\displaystyle p}
7261:{\displaystyle F}
7241:{\displaystyle F}
7035:{\displaystyle k}
6927:{\displaystyle m}
6880:{\displaystyle F}
6857:{\displaystyle M}
6837:{\displaystyle s}
6812:
6684:
6595:
6549:
6539:
6443:
6397:
6387:
6305:{\displaystyle S}
6285:{\displaystyle E}
6088:linear block code
5727:{\displaystyle k}
5503:
5495:
5396:{\displaystyle t}
5053:{\displaystyle k}
4876:
4278:. The polynomial
3597:{\displaystyle k}
3366:{\displaystyle m}
3278:
3222:{\displaystyle k}
3136:{\displaystyle A}
3108:{\displaystyle F}
3091:This matrix is a
2725:{\displaystyle F}
2705:{\displaystyle A}
2617:{\displaystyle C}
2404:{\displaystyle F}
2339:different points
2332:{\displaystyle n}
2285:{\displaystyle m}
1602:agree in at most
1595:{\displaystyle k}
1548:
1540:
1388:primitive element
1277:{\displaystyle q}
1257:{\displaystyle F}
1233:{\displaystyle p}
1213:{\displaystyle k}
1118:
1083:{\displaystyle q}
1063:{\displaystyle q}
1043:{\displaystyle F}
985:{\displaystyle k}
962:{\displaystyle n}
939:{\displaystyle q}
915:
914:
882:RS + CC (15, 1/6)
837:Binary Golay code
625:Data transmission
408:Daniel Gorenstein
267:data transmission
224:
223:
68:Reed–Solomon code
61:Linear block code
18:Reed–Solomon code
16:(Redirected from
22576:
22508:
22507:
22485:
22432:
22409:
22391:Sloane, N. J. A.
22385:
22370:
22345:
22325:
22304:
22284:
22282:
22270:
22252:Solomon, Gustave
22243:
22220:
22197:
22184:
22166:
22149:
22139:
22138:
22136:
22131:on June 30, 2014
22130:
22123:
22102:
22101:
22099:
22098:
22092:
22081:
22072:
22066:
22065:
22055:
22033:
22027:
22026:
22025:
22024:
22019:
22001:
21995:
21994:
21974:
21954:
21948:
21947:
21927:
21907:
21901:
21900:
21891:
21882:(6): 1757–1767,
21869:
21863:
21860:
21854:
21848:
21842:
21839:
21833:
21827:
21826:
21825:
21819:
21804:
21795:
21789:
21788:
21786:
21785:
21770:
21764:
21757:
21751:
21750:
21724:
21715:
21706:
21705:
21679:
21673:
21672:
21652:Immink, K. A. S.
21648:
21642:
21641:
21639:
21628:
21619:
21618:
21616:
21592:
21586:
21585:
21557:
21551:
21550:
21534:
21528:
21527:
21499:
21493:
21488:
21471:
21464:
21408:
21399:
21393:
21382:
21369:
21333:
21313:
21258:
21231:
21128:
21127:
21124:
21122:
21121:
21116:
21108:
21107:
21090:
21088:
21087:
21082:
21074:
21073:
21045:
21043:
21042:
21037:
21029:
21028:
21010:
21009:
20990:
20988:
20987:
20982:
20977:
20976:
20959:
20957:
20956:
20951:
20946:
20945:
20926:
20921:
20903:
20902:
20865:
20856:
20850:
20846:
20840:
20804:
20784:
20756:
20754:
20753:
20748:
20746:
20745:
20682:
20681:
20674:
20673:
20660:
20659:
20646:
20645:
20632:
20631:
20618:
20617:
20604:
20603:
20590:
20589:
20572:
20571:
20286:
20284:
20283:
20278:
20276:
20275:
20212:
20211:
20204:
20203:
20190:
20189:
20176:
20175:
20162:
20161:
20148:
20147:
20134:
20133:
20120:
20119:
20102:
20101:
19823:
19821:
19820:
19815:
19812:
19807:
19798:
19797:
19778:
19773:
19764:
19763:
19750:
19745:
19736:
19735:
19722:
19717:
19708:
19707:
19695:
19694:
19685:
19684:
19672:
19671:
19653:
19652:
19643:
19642:
19630:
19629:
19617:
19616:
19592:
19590:
19589:
19584:
19573:
19572:
19551:
19550:
19535:
19534:
19514:
19495:
19487:The codeword is
19484:
19461:
19431:
19429:
19428:
19423:
19411:
19409:
19408:
19403:
19398:
19397:
19362:
19360:
19359:
19354:
19330:
19328:
19327:
19322:
19320:
19318:
19292:
19284:
19279:
19278:
19277:
19264:
19245:
19243:
19242:
19237:
19235:
19234:
19218:
19216:
19215:
19210:
19208:
19207:
19171:
19168:
19165:
19162:
19159:
19156:
19153:
19150:
19147:
19144:
19141:
19138:
19135:
19132:
19129:
19126:
19122:
19119:
19116:
19112:
19109:
19106:
19103:
19100:
19097:
19094:
19091:
19088:
19085:
19082:
19079:
19076:
19073:
19070:
19067:
19064:
19061:
19058:
19055:
19052:
19049:
19046:
19043:
19040:
19037:
19034:
19031:
19028:
19025:
19022:
19019:
19016:
19012:
19009:
19006:
19002:
18999:
18996:
18993:
18990:
18987:
18984:
18981:
18978:
18975:
18972:
18969:
18966:
18963:
18960:
18957:
18954:
18951:
18948:
18945:
18942:
18939:
18936:
18933:
18930:
18927:
18924:
18921:
18918:
18915:
18912:
18909:
18906:
18903:
18900:
18897:
18894:
18891:
18888:
18885:
18882:
18879:
18876:
18873:
18870:
18867:
18864:
18861:
18858:
18855:
18852:
18849:
18846:
18843:
18840:
18837:
18834:
18831:
18828:
18825:
18822:
18819:
18816:
18813:
18810:
18807:
18804:
18801:
18798:
18795:
18792:
18789:
18786:
18783:
18780:
18777:
18774:
18771:
18768:
18765:
18762:
18759:
18756:
18753:
18750:
18747:
18744:
18741:
18738:
18735:
18732:
18729:
18726:
18723:
18720:
18717:
18714:
18711:
18708:
18705:
18702:
18699:
18696:
18693:
18690:
18687:
18684:
18681:
18678:
18675:
18672:
18669:
18666:
18663:
18660:
18657:
18654:
18651:
18648:
18645:
18642:
18639:
18636:
18633:
18630:
18627:
18624:
18621:
18618:
18615:
18612:
18609:
18606:
18603:
18600:
18597:
18594:
18591:
18588:
18585:
18582:
18579:
18576:
18573:
18570:
18567:
18564:
18561:
18558:
18555:
18552:
18549:
18546:
18543:
18540:
18537:
18534:
18531:
18528:
18525:
18522:
18519:
18516:
18513:
18510:
18507:
18504:
18501:
18498:
18495:
18492:
18489:
18486:
18483:
18480:
18477:
18474:
18471:
18468:
18465:
18462:
18459:
18456:
18453:
18450:
18447:
18444:
18441:
18438:
18435:
18432:
18429:
18426:
18423:
18420:
18417:
18414:
18411:
18408:
18405:
18402:
18399:
18396:
18393:
18390:
18387:
18384:
18381:
18378:
18375:
18372:
18369:
18366:
18363:
18360:
18357:
18354:
18351:
18348:
18345:
18342:
18339:
18336:
18333:
18330:
18327:
18324:
18321:
18318:
18315:
18312:
18309:
18306:
18303:
18300:
18297:
18294:
18291:
18288:
18285:
18282:
18279:
18276:
18273:
18270:
18267:
18264:
18261:
18258:
18255:
18252:
18249:
18246:
18243:
18240:
18237:
18234:
18231:
18228:
18225:
18222:
18219:
18216:
18213:
18210:
18207:
18204:
18201:
18198:
18195:
18192:
18189:
18186:
18185:% Add some zeros
18183:
18180:
18177:
18174:
18171:
18168:
18165:
18162:
18159:
18156:
18153:
18150:
18147:
18144:
18141:
18138:
18135:
18132:
18129:
18126:
18123:
18120:
18117:
18114:
18111:
18108:
18105:
18102:
18099:
18096:
18093:
18090:
18087:
18084:
18081:
18078:
18075:
18072:
18069:
18066:
18063:
18060:
18057:
18054:
18051:
18048:
18045:
18042:
18039:
18036:
18033:
18030:
18027:
18024:
18021:
18018:
18015:
18012:
18009:
18006:
18003:
18000:
17997:
17994:
17991:
17988:
17985:
17982:
17979:
17976:
17973:
17970:
17967:
17964:
17961:
17958:
17955:
17952:
17949:
17946:
17943:
17940:
17937:
17934:
17931:
17928:
17925:
17922:
17919:
17916:
17913:
17910:
17907:
17904:
17901:
17898:
17895:
17892:
17889:
17886:
17883:
17880:
17877:
17874:
17871:
17868:
17865:
17862:
17859:
17856:
17853:
17850:
17847:
17844:
17841:
17838:
17835:
17832:
17829:
17826:
17823:
17820:
17817:
17814:
17811:
17808:
17805:
17802:
17799:
17796:
17793:
17790:
17787:
17784:
17781:
17778:
17775:
17772:
17769:
17766:
17763:
17760:
17757:
17754:
17751:
17748:
17745:
17742:
17739:
17736:
17733:
17730:
17727:
17724:
17721:
17718:
17715:
17712:
17709:
17706:
17703:
17700:
17697:
17694:
17691:
17688:
17685:
17682:
17679:
17676:
17673:
17670:
17667:
17664:
17661:
17657:
17654:
17651:
17636:
17633:
17630:
17627:
17624:
17621:
17618:
17615:
17612:
17609:
17606:
17603:
17600:
17597:
17594:
17591:
17588:
17585:
17582:
17579:
17576:
17573:
17570:
17567:
17564:
17561:
17558:
17555:
17551:
17548:
17545:
17541:
17538:
17535:
17532:
17529:
17526:
17523:
17520:
17517:
17514:
17511:
17508:
17505:
17502:
17499:
17496:
17493:
17490:
17487:
17484:
17481:
17478:
17475:
17472:
17469:
17466:
17463:
17460:
17457:
17454:
17451:
17448:
17445:
17442:
17439:
17436:
17433:
17430:
17427:
17424:
17421:
17418:
17415:
17412:
17409:
17406:
17403:
17400:
17397:
17394:
17391:
17388:
17385:
17382:
17379:
17376:
17373:
17370:
17367:
17364:
17360:
17357:
17354:
17350:
17292:
17283:capacity (up to
17275:
17273:
17272:
17267:
17262:
17261:
17216:
17204:
17192:
17115:
17113:
17112:
17107:
17105:
17086:
17083:
17076:
17075:
17060:
17059:
17041:
17040:
17025:
17024:
17012:
17011:
16996:
16995:
16973:
16972:
16956:
16955:
16946:
16945:
16921:
16920:
16905:
16904:
16892:
16891:
16879:
16877:
16876:
16864:
16852:
16851:
16831:
16829:
16828:
16823:
16821:
16820:
16804:
16802:
16801:
16796:
16794:
16793:
16762:
16760:
16759:
16754:
16752:
16751:
16735:
16733:
16732:
16727:
16725:
16724:
16706:
16704:
16703:
16698:
16681:
16678:
16672:
16671:
16653:
16652:
16640:
16639:
16627:
16626:
16595:coefficients of
16276:
16275:
15859:
15857:
15856:
15851:
15849:
15848:
15841:
15840:
15824:
15823:
15810:
15809:
15800:
15799:
15765:
15764:
15755:
15754:
15741:
15740:
15731:
15730:
15717:
15716:
15707:
15706:
15686:
15685:
15678:
15677:
15659:
15658:
15649:
15648:
15636:
15635:
15622:
15621:
15610:
15609:
15600:
15599:
15587:
15586:
15577:
15576:
15564:
15563:
15550:
15549:
15538:
15537:
15528:
15527:
15515:
15514:
15505:
15504:
15492:
15491:
15482:
15481:
15469:
15468:
15455:
15454:
15443:
15442:
15433:
15432:
15420:
15419:
15410:
15409:
15397:
15396:
15387:
15386:
15374:
15373:
15360:
15359:
15348:
15347:
15338:
15337:
15325:
15324:
15315:
15314:
15302:
15301:
15292:
15291:
15279:
15278:
15265:
15264:
15253:
15252:
15243:
15242:
15230:
15229:
15220:
15219:
15207:
15206:
15197:
15196:
15183:
15182:
15171:
15170:
15161:
15160:
15148:
15147:
15138:
15137:
15124:
15123:
15112:
15111:
15102:
15101:
15067:
15065:
15064:
15059:
15042:
15041:
14984:
14982:
14981:
14976:
14974:
14970:
14969:
14954:
14953:
14935:
14934:
14919:
14918:
14900:
14899:
14890:
14889:
14848:
14847:
14829:
14828:
14813:
14812:
14794:
14793:
14784:
14783:
14751:
14750:
14735:
14734:
14716:
14715:
14700:
14699:
14681:
14680:
14665:
14664:
14408:
14407:
14372:
14370:
14369:
14364:
14362:
14361:
14344:
14343:
14342:
14324:
14323:
14306:
14305:
14304:
14292:
14291:
14237:
14235:
14234:
14229:
14227:
14226:
14211:
14210:
14195:
14194:
14185:
14184:
14172:
14171:
14162:
14161:
14105:Forney algorithm
14074:
14072:
14071:
14066:
14064:
14063:
14035:
14034:
14027:
14026:
14013:
14012:
13995:
13994:
13946:
13944:
13943:
13938:
13936:
13935:
13907:
13906:
13872:
13871:
13864:
13863:
13850:
13849:
13832:
13831:
13790:
13788:
13787:
13782:
13771:
13770:
13752:
13751:
13729:
13728:
13710:
13709:
13687:
13686:
13668:
13667:
13652:
13650:
13649:
13644:
13618:
13617:
13599:
13598:
13580:
13579:
13561:
13560:
13542:
13541:
13520:
13519:
13501:
13500:
13476:
13474:
13473:
13468:
13451:
13450:
13435:
13434:
13419:
13418:
13403:
13402:
13387:
13386:
13322:
13320:
13319:
13314:
13297:
13296:
13281:
13280:
13265:
13264:
13249:
13248:
13233:
13232:
13208:
13207:
13195:
13194:
13151:
13149:
13148:
13143:
13126:
13125:
13110:
13109:
13085:
13084:
13075:
13074:
13040:
13039:
13023:
13004:
13002:
13001:
12996:
12979:
12978:
12963:
12962:
12947:
12946:
12931:
12930:
12909:
12908:
12887:
12886:
12827:
12820:
12813:
12742:
12740:
12739:
12734:
12706:Forney algorithm
12673:
12671:
12670:
12665:
12662:
12654:
12638:
12636:
12635:
12630:
12628:
12627:
12569:
12567:
12566:
12561:
12559:
12558:
12551:
12550:
12521:
12520:
12498:
12497:
12468:
12467:
12460:
12459:
12439:
12438:
12419:
12418:
12401:
12400:
12393:
12392:
12367:
12366:
12349:
12348:
12313:
12312:
12290:
12289:
12278:
12277:
12264:
12263:
12247:
12246:
12235:
12234:
12190:
12188:
12187:
12182:
12180:
12179:
12158:
12157:
12148:
12147:
12117:
12116:
12101:
12100:
12082:
12081:
12072:
12071:
12055:
12053:
12052:
12047:
12045:
12044:
12020:
12018:
12017:
12012:
12004:
12003:
11994:
11993:
11981:
11980:
11965:
11964:
11940:
11939:
11918:
11917:
11905:
11904:
11880:
11878:
11877:
11872:
11864:
11860:
11858:
11853:
11844:
11843:
11833:
11828:
11808:
11807:
11789:
11785:
11783:
11766:
11757:
11756:
11746:
11741:
11721:
11720:
11708:
11704:
11702:
11685:
11676:
11675:
11665:
11660:
11640:
11639:
11627:
11623:
11621:
11610:
11601:
11600:
11590:
11585:
11558:
11556:
11555:
11550:
11536:
11534:
11533:
11528:
11520:
11516:
11514:
11509:
11500:
11499:
11490:
11489:
11479:
11474:
11445:
11441:
11439:
11422:
11413:
11412:
11403:
11402:
11392:
11387:
11364:
11360:
11358:
11341:
11332:
11331:
11322:
11321:
11311:
11306:
11283:
11279:
11277:
11266:
11257:
11256:
11246:
11241:
11212:
11210:
11209:
11204:
11196:
11192:
11190:
11185:
11176:
11175:
11166:
11165:
11146:
11129:
11120:
11119:
11110:
11109:
11096:
11079:
11070:
11069:
11060:
11059:
11046:
11035:
11026:
11025:
11010:
11005:
10973:
10971:
10970:
10965:
10963:
10952:
10947:
10938:
10937:
10928:
10927:
10908:
10891:
10882:
10881:
10872:
10871:
10858:
10841:
10832:
10831:
10822:
10821:
10808:
10797:
10788:
10787:
10777:
10766:
10758:
10748:
10737:
10728:
10727:
10718:
10717:
10698:
10690:
10680:
10669:
10660:
10659:
10650:
10649:
10636:
10628:
10618:
10607:
10598:
10597:
10588:
10587:
10574:
10563:
10554:
10553:
10543:
10533:
10529:
10527:
10519:
10510:
10509:
10490:
10482:
10473:
10472:
10459:
10451:
10442:
10441:
10420:
10409:
10400:
10399:
10389:
10375:
10367:
10351:
10340:
10331:
10330:
10320:
10309:
10307:
10306:
10301:
10298:
10287:
10278:
10277:
10261:
10259:
10258:
10253:
10229:
10227:
10226:
10221:
10207:
10205:
10204:
10199:
10187:
10179:
10157:
10155:
10154:
10149:
10126:
10125:
10112:
10104:
10079:
10077:
10076:
10071:
10068:
10060:
10038:
10036:
10035:
10030:
10027:
10019:
10003:
9993:
9991:
9990:
9985:
9983:
9982:
9973:
9972:
9954:
9953:
9944:
9943:
9931:
9930:
9921:
9920:
9899:
9898:
9876:
9871:
9834:
9805:
9803:
9802:
9797:
9785:
9783:
9782:
9777:
9759:
9757:
9756:
9751:
9705:
9703:
9702:
9697:
9695:
9694:
9687:
9686:
9660:
9659:
9646:
9645:
9625:
9624:
9617:
9616:
9596:
9595:
9582:
9581:
9564:
9563:
9555:
9544:
9527:
9516:
9504:
9493:
9457:
9452:
9435:
9430:
9418:
9413:
9399:
9394:
9377:
9372:
9360:
9355:
9306:
9289:
9287:
9286:
9281:
9278:
9273:
9261:
9260:
9251:
9250:
9249:
9248:
9228:
9227:
9226:
9225:
9202:
9201:
9200:
9199:
9185:
9184:
9163:
9161:
9160:
9155:
9152:
9147:
9138:
9137:
9127:
9122:
9104:
9103:
9085:
9083:
9082:
9077:
9075:
9074:
9073:
9072:
9055:
9054:
9043:
9039:
9038:
9037:
9036:
9019:
9018:
8970:
8968:
8967:
8962:
8960:
8959:
8943:
8941:
8940:
8935:
8914:
8912:
8911:
8906:
8895:
8894:
8872:
8870:
8869:
8864:
8862:
8821:
8820:
8819:
8818:
8808:
8804:
8803:
8789:
8788:
8787:
8786:
8771:
8766:
8745:
8738:
8737:
8716:
8709:
8708:
8681:
8680:
8659:
8658:
8637:
8636:
8614:
8613:
8584:
8582:
8581:
8576:
8574:
8573:
8555:
8554:
8483:
8481:
8480:
8475:
8473:
8472:
8471:
8470:
8456:
8455:
8454:
8453:
8438:
8433:
8368:
8366:
8365:
8360:
8358:
8357:
8348:
8347:
8337:
8326:
8290:
8288:
8287:
8282:
8213:
8211:
8210:
8205:
8168:
8155:
8154:
8132:
8130:
8129:
8124:
8113:
8112:
8097:
8096:
8069:
8068:
8053:
8052:
8001:
7999:
7998:
7993:
7985:
7984:
7965:
7954:
7899:
7897:
7896:
7891:
7889:
7888:
7879:
7878:
7868:
7857:
7812:
7810:
7809:
7804:
7799:
7798:
7774:
7773:
7755:
7754:
7668:
7666:
7665:
7660:
7644:
7642:
7641:
7636:
7634:
7633:
7621:
7620:
7604:
7602:
7601:
7596:
7588:
7587:
7560:
7559:
7534:
7532:
7531:
7526:
7496:
7494:
7493:
7488:
7480:
7479:
7446:
7445:
7420:
7418:
7417:
7412:
7400:
7398:
7397:
7392:
7380:
7378:
7377:
7372:
7370:
7362:
7344:
7342:
7341:
7336:
7294:
7292:
7291:
7286:
7284:
7283:
7267:
7265:
7264:
7259:
7247:
7245:
7244:
7239:
7223:
7221:
7220:
7215:
7173:
7171:
7170:
7165:
7123:
7121:
7120:
7115:
7097:
7095:
7094:
7089:
7067:
7065:
7064:
7059:
7041:
7039:
7038:
7033:
7017:
7015:
7014:
7009:
6995:
6994:
6972:
6970:
6969:
6964:
6956:
6955:
6933:
6931:
6930:
6925:
6913:
6911:
6910:
6905:
6903:
6902:
6886:
6884:
6883:
6878:
6863:
6861:
6860:
6855:
6843:
6841:
6840:
6835:
6823:
6821:
6820:
6815:
6813:
6811:
6804:
6803:
6790:
6775:
6773:
6772:
6767:
6765:
6764:
6734:
6733:
6717:
6715:
6714:
6709:
6698:
6697:
6685:
6677:
6662:
6660:
6659:
6654:
6652:
6651:
6636:
6635:
6616:
6611:
6602:
6601:
6600:
6587:
6576:
6571:
6550:
6542:
6540:
6532:
6527:
6526:
6510:
6508:
6507:
6502:
6500:
6499:
6484:
6483:
6464:
6459:
6450:
6449:
6448:
6435:
6424:
6419:
6398:
6390:
6388:
6386:
6379:
6378:
6368:
6367:
6352:
6347:
6346:
6311:
6309:
6308:
6303:
6291:
6289:
6288:
6283:
6271:
6237:
6235:
6234:
6229:
6224:
6197:
6195:
6194:
6189:
6152:
6150:
6149:
6144:
6124:
6123:
6107:Hamming distance
6078:
6076:
6075:
6070:
6031:
6030:
6009:
6008:
5987:
5986:
5974:
5973:
5927:
5925:
5924:
5919:
5898:
5896:
5895:
5890:
5888:
5876:
5874:
5873:
5868:
5845:
5843:
5842:
5837:
5822:
5821:
5809:
5808:
5762:
5760:
5759:
5754:
5733:
5731:
5730:
5725:
5713:
5711:
5710:
5705:
5684:
5682:
5681:
5676:
5674:
5673:
5642:
5640:
5639:
5634:
5632:
5631:
5619:
5618:
5600:
5599:
5581:
5580:
5558:
5556:
5555:
5550:
5530:
5528:
5527:
5522:
5505:
5504:
5501:
5493:
5492:
5491:
5455:
5454:
5438:
5436:
5435:
5430:
5419:
5418:
5402:
5400:
5399:
5394:
5382:
5380:
5379:
5374:
5353:
5351:
5350:
5345:
5321:
5319:
5318:
5313:
5311:
5310:
5294:
5292:
5291:
5286:
5262:
5260:
5259:
5254:
5233:
5231:
5230:
5225:
5204:
5202:
5201:
5196:
5175:
5173:
5172:
5167:
5146:
5144:
5143:
5138:
5117:
5115:
5114:
5109:
5088:
5086:
5085:
5080:
5059:
5057:
5056:
5051:
5039:
5037:
5036:
5031:
5010:
5008:
5007:
5002:
4943:
4941:
4940:
4935:
4930:
4926:
4925:
4924:
4900:
4899:
4887:
4886:
4877:
4874:
4872:
4871:
4862:
4861:
4851:
4846:
4815:
4814:
4807:
4803:
4802:
4801:
4783:
4782:
4770:
4769:
4747:
4733:
4731:
4730:
4725:
4705:
4703:
4702:
4697:
4695:
4694:
4676:
4675:
4654:
4653:
4620:
4619:
4607:
4606:
4594:
4590:
4589:
4588:
4547:
4543:
4542:
4541:
4515:
4511:
4510:
4509:
4468:
4466:
4465:
4460:
4448:
4446:
4445:
4440:
4419:
4417:
4416:
4411:
4393:
4391:
4390:
4385:
4361:
4359:
4358:
4353:
4335:
4333:
4332:
4327:
4306:
4304:
4303:
4298:
4277:
4275:
4274:
4269:
4248:
4246:
4245:
4240:
4216:
4214:
4213:
4208:
4190:
4188:
4187:
4182:
4161:
4159:
4158:
4153:
4117:
4115:
4114:
4109:
4104:
4103:
4079:
4078:
4051:
4050:
4026:
4025:
4010:
4008:
4007:
4002:
4000:
3999:
3987:
3986:
3952:
3950:
3949:
3944:
3942:
3941:
3931:
3930:
3912:
3911:
3888:
3887:
3875:
3874:
3858:
3857:
3845:
3844:
3805:
3803:
3802:
3797:
3795:
3794:
3766:
3764:
3763:
3758:
3756:
3755:
3739:
3737:
3736:
3731:
3729:
3728:
3704:
3703:
3687:
3685:
3684:
3679:
3674:
3673:
3655:
3654:
3633:
3632:
3620:
3619:
3603:
3601:
3600:
3595:
3584:since the first
3580:This variant is
3577:
3575:
3574:
3569:
3567:
3566:
3556:
3555:
3537:
3536:
3513:
3512:
3500:
3499:
3483:
3482:
3470:
3469:
3428:
3426:
3425:
3420:
3418:
3417:
3393:
3392:
3372:
3370:
3369:
3364:
3352:
3350:
3349:
3344:
3342:
3341:
3325:
3323:
3322:
3317:
3279:
3276:
3274:
3273:
3258:
3257:
3245:
3244:
3228:
3226:
3225:
3220:
3208:
3206:
3205:
3200:
3198:
3197:
3181:
3179:
3178:
3173:
3171:
3170:
3142:
3140:
3139:
3134:
3121:generator matrix
3114:
3112:
3111:
3106:
3088:
3086:
3085:
3080:
3078:
3077:
3070:
3069:
3043:
3042:
3029:
3028:
3011:
3010:
3002:
2991:
2968:
2963:
2946:
2945:
2893:
2882:
2865:
2860:
2849:
2848:
2829:
2818:
2801:
2796:
2785:
2784:
2731:
2729:
2728:
2723:
2711:
2709:
2708:
2703:
2691:
2689:
2688:
2683:
2665:
2663:
2662:
2657:
2623:
2621:
2620:
2615:
2603:
2601:
2600:
2595:
2593:
2592:
2582:
2581:
2563:
2562:
2539:
2538:
2526:
2525:
2509:
2508:
2496:
2495:
2456:
2454:
2453:
2448:
2446:
2445:
2433:
2432:
2410:
2408:
2407:
2402:
2390:
2388:
2387:
2382:
2380:
2379:
2355:
2354:
2338:
2336:
2335:
2330:
2318:
2316:
2315:
2310:
2308:
2307:
2291:
2289:
2288:
2283:
2272:The codeword of
2271:
2269:
2268:
2263:
2257:
2256:
2247:
2246:
2236:
2225:
2198:
2197:
2181:
2179:
2178:
2173:
2171:
2170:
2154:
2152:
2151:
2146:
2144:
2143:
2128:
2127:
2103:
2102:
2053:
2051:
2050:
2045:
2043:
2042:
2024:
2023:
1983:
1981:
1980:
1975:
1973:
1972:
1945:
1943:
1942:
1937:
1932:
1891:
1889:
1888:
1883:
1878:
1857:
1855:
1854:
1849:
1832:
1806:
1792:
1772:
1751:
1749:
1748:
1743:
1709:
1707:
1706:
1701:
1683:
1681:
1680:
1675:
1627:
1625:
1624:
1619:
1601:
1599:
1598:
1593:
1577:
1575:
1574:
1569:
1563:
1562:
1549:
1546:
1541:
1538:
1532:
1531:
1524:
1523:
1514:
1513:
1486:
1485:
1464:
1463:
1448:
1447:
1440:
1439:
1430:
1418:
1416:
1415:
1410:
1408:
1341:
1339:
1338:
1333:
1331:
1330:
1312:
1311:
1296:distinct points
1288:is evaluated at
1283:
1281:
1280:
1275:
1263:
1261:
1260:
1255:
1239:
1237:
1236:
1231:
1219:
1217:
1216:
1211:
1187:
1185:
1184:
1179:
1155:
1153:
1152:
1147:
1129:
1127:
1126:
1121:
1119:
1111:
1089:
1087:
1086:
1081:
1069:
1067:
1066:
1061:
1049:
1047:
1046:
1041:
1026:
1024:
1023:
1018:
991:
989:
988:
983:
968:
966:
965:
960:
945:
943:
942:
937:
828:Mariner, Viking
823:Reed-Muller code
780:
776:
765:Shannon capacity
738:Viterbi decoders
639:erasure channels
354:
351:. The choice of
350:
346:
335:
327:
315:
311:
307:
303:
216:
209:
202:
170:Berlekamp–Massey
30:
29:
21:
22584:
22583:
22579:
22578:
22577:
22575:
22574:
22573:
22554:
22553:
22515:
22513:Implementations
22505:
22444:
22439:
22430:
22414:Reed, Irving S.
22407:
22350:Forney, Jr., G.
22323:
22283:, Lecture Notes
22280:
22268:10.1137/0108018
22248:Reed, Irving S.
22195:
22182:
22147:
22134:
22132:
22128:
22121:
22111:
22109:Further reading
22106:
22105:
22096:
22094:
22090:
22079:
22073:
22069:
22034:
22030:
22022:
22020:
22002:
21998:
21991:
21955:
21951:
21944:
21908:
21904:
21870:
21866:
21861:
21857:
21849:
21845:
21840:
21836:
21823:
21821:
21817:
21802:
21796:
21792:
21783:
21781:
21772:
21771:
21767:
21758:
21754:
21733:(11): 2142–56.
21722:
21716:
21709:
21694:
21680:
21676:
21670:
21649:
21645:
21637:
21629:
21622:
21593:
21589:
21574:
21558:
21554:
21535:
21531:
21516:10.1137/0109020
21500:
21496:
21489:
21485:
21480:
21475:
21474:
21465:
21461:
21456:
21414:
21409:
21403:
21395:
21384:
21373:
21370:
21336:
21334:
21316:
21314:
21288:
21259:
21248:
21234:
21232:
21213:
21199:
21149:
21141:
21103:
21099:
21097:
21094:
21093:
21066:
21062:
21060:
21057:
21056:
21024:
21020:
21005:
21001:
20996:
20993:
20992:
20972:
20968:
20966:
20963:
20962:
20941:
20937:
20922:
20911:
20895:
20891:
20889:
20886:
20885:
20879:
20871:
20866:
20860:
20852:
20848:
20844:
20841:
20807:
20805:
20787:
20785:
20759:
20740:
20739:
20733:
20732:
20726:
20725:
20719:
20718:
20712:
20711:
20705:
20704:
20698:
20697:
20687:
20686:
20676:
20675:
20669:
20665:
20662:
20661:
20655:
20651:
20648:
20647:
20641:
20637:
20634:
20633:
20627:
20623:
20620:
20619:
20613:
20609:
20606:
20605:
20599:
20595:
20592:
20591:
20585:
20581:
20574:
20573:
20566:
20565:
20560:
20555:
20550:
20545:
20540:
20535:
20529:
20528:
20523:
20518:
20513:
20508:
20503:
20498:
20492:
20491:
20486:
20481:
20476:
20471:
20466:
20461:
20455:
20454:
20449:
20444:
20439:
20434:
20429:
20424:
20418:
20417:
20412:
20407:
20402:
20397:
20392:
20387:
20381:
20380:
20375:
20370:
20365:
20360:
20355:
20350:
20344:
20343:
20338:
20333:
20328:
20323:
20318:
20313:
20303:
20302:
20300:
20297:
20296:
20270:
20269:
20263:
20262:
20256:
20255:
20249:
20248:
20242:
20241:
20235:
20234:
20228:
20227:
20217:
20216:
20206:
20205:
20199:
20195:
20192:
20191:
20185:
20181:
20178:
20177:
20171:
20167:
20164:
20163:
20157:
20153:
20150:
20149:
20143:
20139:
20136:
20135:
20129:
20125:
20122:
20121:
20115:
20111:
20104:
20103:
20096:
20095:
20090:
20085:
20080:
20075:
20070:
20065:
20059:
20058:
20053:
20048:
20043:
20038:
20033:
20028:
20022:
20021:
20016:
20011:
20006:
20001:
19996:
19991:
19985:
19984:
19979:
19974:
19969:
19964:
19959:
19954:
19948:
19947:
19942:
19937:
19932:
19927:
19922:
19917:
19911:
19910:
19905:
19900:
19895:
19890:
19885:
19880:
19874:
19873:
19868:
19863:
19858:
19853:
19848:
19843:
19833:
19832:
19830:
19827:
19826:
19808:
19803:
19793:
19789:
19774:
19769:
19759:
19755:
19746:
19741:
19731:
19727:
19718:
19713:
19703:
19699:
19690:
19686:
19680:
19676:
19667:
19663:
19648:
19644:
19638:
19634:
19625:
19621:
19612:
19608:
19606:
19603:
19602:
19568:
19564:
19546:
19542:
19530:
19526:
19524:
19521:
19520:
19515:
19501:
19496:
19490:
19485:
19467:
19462:
19456:
19448:
19438:
19417:
19414:
19413:
19393:
19389:
19381:
19378:
19377:
19369:
19336:
19333:
19332:
19293:
19285:
19283:
19273:
19260:
19259:
19258:
19256:
19253:
19252:
19230:
19226:
19224:
19221:
19220:
19203:
19199:
19197:
19194:
19193:
19187:
19178:
19173:
19172:
19169:
19166:
19163:
19160:
19157:
19154:
19151:
19148:
19145:
19142:
19139:
19136:
19133:
19130:
19127:
19124:
19120:
19117:
19114:
19110:
19107:
19104:
19101:
19098:
19095:
19092:
19089:
19086:
19083:
19080:
19077:
19074:
19071:
19068:
19065:
19062:
19059:
19056:
19053:
19050:
19047:
19044:
19041:
19038:
19035:
19032:
19029:
19026:
19023:
19020:
19017:
19014:
19010:
19007:
19004:
19000:
18997:
18994:
18991:
18988:
18985:
18982:
18979:
18976:
18973:
18970:
18967:
18964:
18961:
18958:
18955:
18952:
18949:
18946:
18943:
18940:
18937:
18934:
18931:
18928:
18925:
18922:
18919:
18916:
18913:
18910:
18907:
18904:
18901:
18898:
18895:
18892:
18889:
18886:
18883:
18880:
18877:
18874:
18871:
18868:
18865:
18862:
18859:
18856:
18853:
18850:
18847:
18844:
18841:
18838:
18835:
18832:
18829:
18826:
18823:
18820:
18817:
18814:
18811:
18808:
18805:
18802:
18799:
18796:
18793:
18790:
18787:
18784:
18781:
18778:
18775:
18772:
18769:
18766:
18763:
18760:
18757:
18754:
18751:
18748:
18745:
18742:
18739:
18736:
18733:
18730:
18727:
18724:
18721:
18718:
18715:
18712:
18709:
18706:
18703:
18700:
18697:
18694:
18691:
18688:
18685:
18682:
18679:
18676:
18673:
18670:
18667:
18664:
18661:
18658:
18655:
18652:
18649:
18646:
18643:
18640:
18637:
18634:
18631:
18628:
18625:
18622:
18619:
18616:
18613:
18610:
18607:
18604:
18601:
18598:
18595:
18592:
18589:
18586:
18583:
18580:
18577:
18574:
18571:
18568:
18565:
18562:
18559:
18556:
18553:
18550:
18547:
18544:
18541:
18538:
18535:
18532:
18529:
18526:
18523:
18520:
18517:
18514:
18511:
18508:
18505:
18502:
18499:
18496:
18493:
18490:
18487:
18484:
18481:
18478:
18475:
18472:
18469:
18466:
18463:
18460:
18457:
18454:
18451:
18448:
18445:
18442:
18439:
18436:
18433:
18430:
18427:
18424:
18421:
18418:
18415:
18412:
18409:
18406:
18403:
18400:
18397:
18394:
18391:
18388:
18385:
18382:
18379:
18376:
18373:
18370:
18367:
18364:
18361:
18358:
18355:
18352:
18349:
18346:
18343:
18340:
18337:
18334:
18331:
18328:
18325:
18322:
18319:
18316:
18313:
18310:
18307:
18304:
18301:
18298:
18295:
18292:
18289:
18286:
18283:
18280:
18277:
18274:
18271:
18268:
18265:
18262:
18259:
18256:
18253:
18250:
18247:
18244:
18241:
18238:
18235:
18232:
18229:
18226:
18223:
18220:
18217:
18214:
18211:
18208:
18205:
18202:
18199:
18196:
18193:
18190:
18187:
18184:
18181:
18178:
18175:
18172:
18169:
18166:
18163:
18160:
18157:
18154:
18151:
18148:
18145:
18142:
18139:
18136:
18133:
18130:
18127:
18124:
18121:
18118:
18115:
18112:
18109:
18106:
18103:
18100:
18097:
18094:
18091:
18088:
18085:
18082:
18079:
18076:
18073:
18070:
18067:
18064:
18061:
18058:
18055:
18052:
18049:
18046:
18043:
18040:
18037:
18034:
18031:
18028:
18025:
18022:
18019:
18016:
18013:
18010:
18007:
18004:
18001:
17998:
17995:
17992:
17989:
17986:
17983:
17980:
17977:
17974:
17971:
17968:
17965:
17962:
17959:
17956:
17953:
17950:
17947:
17944:
17941:
17938:
17935:
17932:
17929:
17926:
17923:
17920:
17917:
17914:
17911:
17908:
17905:
17902:
17899:
17896:
17893:
17890:
17887:
17884:
17881:
17878:
17875:
17872:
17869:
17866:
17863:
17860:
17857:
17854:
17851:
17848:
17845:
17842:
17839:
17836:
17833:
17830:
17827:
17824:
17821:
17818:
17815:
17812:
17809:
17806:
17803:
17800:
17797:
17794:
17791:
17788:
17785:
17782:
17779:
17776:
17773:
17770:
17768:% Get the alpha
17767:
17764:
17761:
17758:
17755:
17752:
17749:
17746:
17743:
17740:
17737:
17734:
17731:
17728:
17725:
17722:
17719:
17716:
17713:
17710:
17707:
17704:
17701:
17698:
17695:
17692:
17689:
17686:
17683:
17680:
17677:
17674:
17671:
17668:
17665:
17662:
17659:
17655:
17652:
17649:
17643:
17638:
17637:
17634:
17631:
17628:
17625:
17622:
17619:
17616:
17613:
17610:
17607:
17604:
17601:
17598:
17595:
17592:
17589:
17586:
17583:
17580:
17577:
17574:
17571:
17568:
17565:
17562:
17559:
17556:
17553:
17549:
17546:
17543:
17539:
17536:
17533:
17530:
17527:
17524:
17521:
17518:
17515:
17512:
17509:
17506:
17503:
17500:
17497:
17494:
17491:
17488:
17485:
17482:
17479:
17476:
17473:
17470:
17467:
17464:
17461:
17458:
17455:
17452:
17449:
17446:
17443:
17440:
17437:
17434:
17431:
17428:
17425:
17422:
17419:
17416:
17413:
17410:
17407:
17404:
17401:
17398:
17395:
17392:
17389:
17387:% Get the alpha
17386:
17383:
17380:
17377:
17374:
17371:
17368:
17365:
17362:
17358:
17355:
17352:
17348:
17338:
17333:
17324:Alexander Vardy
17299:
17284:
17257:
17253:
17242:
17239:
17238:
17206:
17198:
17193:. The distance
17183:
17168:Singleton bound
17164:
17118:Then calculate
17103:
17102:
17082:
17080:
17065:
17061:
17055:
17051:
17030:
17026:
17020:
17016:
17001:
16997:
16991:
16987:
16974:
16968:
16964:
16961:
16960:
16951:
16947:
16935:
16931:
16910:
16906:
16900:
16896:
16887:
16883:
16872:
16868:
16863:
16853:
16847:
16843:
16839:
16837:
16834:
16833:
16816:
16812:
16810:
16807:
16806:
16789:
16785:
16783:
16780:
16779:
16747:
16743:
16741:
16738:
16737:
16720:
16716:
16714:
16711:
16710:
16677:
16667:
16663:
16648:
16644:
16635:
16631:
16622:
16618:
16616:
16613:
16612:
16442:
16437:
16431:
16419:
16409:
16297:
16289:
16271:
16264:
16260:
16251:
16249:
16240:
16226:
16224:
16215:
16199:
16177:
16176:
16163:
16153:
16145:
16132:
16122:
16114:
16104:
16078:
16061:
16054:
16039:
16029:
16004:
15994:
15985:
15975:
15966:
15956:
15937:
15919:
15913:
15896:
15875:
15843:
15842:
15836:
15832:
15829:
15828:
15819:
15815:
15812:
15811:
15805:
15801:
15795:
15791:
15788:
15787:
15781:
15780:
15774:
15773:
15767:
15766:
15760:
15756:
15750:
15746:
15743:
15742:
15736:
15732:
15726:
15722:
15719:
15718:
15712:
15708:
15702:
15698:
15691:
15690:
15680:
15679:
15673:
15669:
15666:
15665:
15660:
15654:
15650:
15644:
15640:
15631:
15627:
15624:
15623:
15617:
15613:
15611:
15605:
15601:
15595:
15591:
15582:
15578:
15572:
15568:
15559:
15555:
15552:
15551:
15545:
15541:
15539:
15533:
15529:
15523:
15519:
15510:
15506:
15500:
15496:
15487:
15483:
15477:
15473:
15464:
15460:
15457:
15456:
15450:
15446:
15444:
15438:
15434:
15428:
15424:
15415:
15411:
15405:
15401:
15392:
15388:
15382:
15378:
15369:
15365:
15362:
15361:
15355:
15351:
15349:
15343:
15339:
15333:
15329:
15320:
15316:
15310:
15306:
15297:
15293:
15287:
15283:
15274:
15270:
15267:
15266:
15260:
15256:
15254:
15248:
15244:
15238:
15234:
15225:
15221:
15215:
15211:
15202:
15198:
15192:
15188:
15185:
15184:
15178:
15174:
15172:
15166:
15162:
15156:
15152:
15143:
15139:
15133:
15129:
15126:
15125:
15119:
15115:
15113:
15107:
15103:
15097:
15093:
15086:
15085:
15083:
15080:
15079:
15037:
15033:
14992:
14989:
14988:
14972:
14971:
14965:
14961:
14949:
14945:
14924:
14920:
14908:
14904:
14895:
14891:
14885:
14881:
14874:
14859:
14858:
14843:
14839:
14818:
14814:
14802:
14798:
14789:
14785:
14779:
14775:
14768:
14753:
14752:
14746:
14742:
14730:
14726:
14705:
14701:
14689:
14685:
14670:
14666:
14660:
14656:
14649:
14633:
14631:
14628:
14627:
14593:
14424:
14403:
14351:
14347:
14338:
14334:
14313:
14309:
14300:
14296:
14287:
14283:
14275:
14272:
14271:
14256:
14222:
14218:
14206:
14202:
14190:
14186:
14180:
14176:
14167:
14163:
14157:
14153:
14151:
14148:
14147:
14144:
14142:
14138:
14134:
14129:
14127:
14123:
14119:
14114:
14111:
14098:
14092:
14090:
14083:
14058:
14057:
14051:
14050:
14040:
14039:
14029:
14028:
14022:
14018:
14015:
14014:
14008:
14004:
13997:
13996:
13989:
13988:
13983:
13977:
13976:
13971:
13961:
13960:
13958:
13955:
13954:
13930:
13929:
13923:
13922:
13912:
13911:
13901:
13900:
13891:
13890:
13877:
13876:
13866:
13865:
13859:
13855:
13852:
13851:
13845:
13841:
13834:
13833:
13826:
13825:
13820:
13814:
13813:
13808:
13798:
13797:
13795:
13792:
13791:
13766:
13762:
13747:
13743:
13724:
13720:
13705:
13701:
13682:
13678:
13663:
13659:
13657:
13654:
13653:
13613:
13609:
13594:
13590:
13575:
13571:
13556:
13552:
13537:
13533:
13515:
13511:
13496:
13492:
13490:
13487:
13486:
13446:
13442:
13430:
13426:
13414:
13410:
13398:
13394:
13382:
13378:
13328:
13325:
13324:
13292:
13288:
13276:
13272:
13260:
13256:
13244:
13240:
13228:
13224:
13203:
13199:
13190:
13186:
13156:
13153:
13152:
13121:
13117:
13105:
13101:
13080:
13076:
13070:
13066:
13035:
13031:
13029:
13026:
13025:
13006:
12974:
12970:
12958:
12954:
12942:
12938:
12926:
12922:
12904:
12900:
12882:
12878:
12837:
12834:
12833:
12822:
12815:
12808:
12805:
12781:
12780:
12771:
12756:
12748:
12728:
12725:
12724:
12721:
12714:
12702:error equations
12698:
12691:
12684:
12655:
12650:
12644:
12641:
12640:
12623:
12619:
12617:
12614:
12613:
12610:
12604:
12597:
12553:
12552:
12540:
12536:
12530:
12529:
12523:
12522:
12510:
12506:
12500:
12499:
12487:
12483:
12473:
12472:
12462:
12461:
12455:
12451:
12448:
12447:
12441:
12440:
12428:
12424:
12421:
12420:
12414:
12410:
12403:
12402:
12395:
12394:
12379:
12375:
12373:
12368:
12356:
12352:
12350:
12344:
12340:
12337:
12336:
12331:
12326:
12321:
12315:
12314:
12302:
12298:
12296:
12291:
12285:
12281:
12279:
12273:
12269:
12266:
12265:
12259:
12255:
12253:
12248:
12242:
12238:
12236:
12230:
12226:
12219:
12218:
12216:
12213:
12212:
12210:
12169:
12165:
12153:
12149:
12131:
12127:
12106:
12102:
12090:
12086:
12077:
12073:
12067:
12063:
12061:
12058:
12057:
12034:
12030:
12028:
12025:
12024:
11999:
11995:
11989:
11985:
11970:
11966:
11954:
11950:
11923:
11919:
11913:
11909:
11894:
11890:
11888:
11885:
11884:
11854:
11849:
11839:
11835:
11829:
11818:
11813:
11809:
11803:
11799:
11767:
11762:
11752:
11748:
11742:
11731:
11726:
11722:
11716:
11712:
11686:
11681:
11671:
11667:
11661:
11650:
11645:
11641:
11635:
11631:
11611:
11606:
11596:
11592:
11586:
11575:
11570:
11566:
11564:
11561:
11560:
11544:
11541:
11540:
11510:
11505:
11495:
11491:
11485:
11481:
11475:
11464:
11459:
11455:
11423:
11418:
11408:
11404:
11398:
11394:
11388:
11377:
11372:
11368:
11342:
11337:
11327:
11323:
11317:
11313:
11307:
11296:
11291:
11287:
11267:
11262:
11252:
11248:
11242:
11231:
11226:
11222:
11220:
11217:
11216:
11186:
11181:
11171:
11167:
11161:
11157:
11130:
11125:
11115:
11111:
11105:
11101:
11080:
11075:
11065:
11061:
11055:
11051:
11036:
11031:
11021:
11017:
11016:
11012:
11006:
10995:
10989:
10986:
10985:
10961:
10960:
10948:
10943:
10933:
10929:
10923:
10919:
10892:
10887:
10877:
10873:
10867:
10863:
10842:
10837:
10827:
10823:
10817:
10813:
10798:
10793:
10783:
10779:
10775:
10774:
10759:
10754:
10738:
10733:
10723:
10719:
10713:
10709:
10691:
10686:
10670:
10665:
10655:
10651:
10645:
10641:
10629:
10624:
10608:
10603:
10593:
10589:
10583:
10579:
10564:
10559:
10549:
10545:
10541:
10540:
10520:
10515:
10505:
10501:
10483:
10478:
10468:
10464:
10452:
10447:
10437:
10433:
10426:
10422:
10410:
10405:
10395:
10391:
10387:
10386:
10368:
10363:
10341:
10336:
10326:
10322:
10317:
10315:
10312:
10311:
10288:
10283:
10273:
10269:
10267:
10264:
10263:
10235:
10232:
10231:
10215:
10212:
10211:
10180:
10175:
10163:
10160:
10159:
10121:
10117:
10105:
10100:
10085:
10082:
10081:
10061:
10056:
10044:
10041:
10040:
10020:
10015:
10009:
10006:
10005:
9997:
9978:
9974:
9968:
9964:
9949:
9945:
9939:
9935:
9926:
9922:
9916:
9912:
9894:
9890:
9872:
9861:
9840:
9837:
9836:
9828:
9819:
9812:
9791:
9788:
9787:
9768:
9765:
9764:
9739:
9736:
9735:
9732:
9719:
9713:
9689:
9688:
9676:
9672:
9669:
9668:
9662:
9661:
9655:
9651:
9648:
9647:
9641:
9637:
9630:
9629:
9619:
9618:
9612:
9608:
9605:
9604:
9598:
9597:
9591:
9587:
9584:
9583:
9577:
9573:
9566:
9565:
9558:
9557:
9545:
9540:
9534:
9529:
9517:
9512:
9506:
9494:
9489:
9482:
9481:
9476:
9471:
9466:
9460:
9459:
9453:
9448:
9442:
9437:
9431:
9426:
9420:
9414:
9409:
9402:
9401:
9395:
9390:
9384:
9379:
9373:
9368:
9362:
9356:
9351:
9340:
9339:
9337:
9334:
9333:
9330:
9323:
9316:
9294:
9274:
9269:
9256:
9252:
9244:
9240:
9239:
9235:
9221:
9217:
9210:
9206:
9195:
9191:
9190:
9186:
9180:
9176:
9171:
9168:
9167:
9148:
9143:
9133:
9129:
9123:
9112:
9099:
9095:
9093:
9090:
9089:
9068:
9064:
9063:
9059:
9050:
9046:
9032:
9028:
9027:
9023:
9014:
9010:
9008:
9005:
9004:
9001:
8991:
8981:
8955:
8951:
8949:
8946:
8945:
8920:
8917:
8916:
8890:
8886:
8878:
8875:
8874:
8860:
8859:
8814:
8810:
8809:
8799:
8795:
8791:
8790:
8782:
8778:
8777:
8773:
8767:
8756:
8743:
8742:
8733:
8729:
8714:
8713:
8704:
8700:
8676:
8672:
8654:
8650:
8632:
8628:
8615:
8609:
8605:
8601:
8599:
8596:
8595:
8593:
8563:
8559:
8550:
8546:
8544:
8541:
8540:
8537:
8507:). From those,
8505:
8504:
8495:
8466:
8462:
8461:
8457:
8449:
8445:
8444:
8440:
8434:
8423:
8402:
8399:
8398:
8391:
8376:
8353:
8349:
8343:
8339:
8327:
8316:
8295:
8292:
8291:
8237:
8234:
8233:
8150:
8146:
8138:
8135:
8134:
8108:
8104:
8092:
8088:
8064:
8060:
8048:
8044:
8030:
8027:
8026:
7980:
7976:
7955:
7944:
7923:
7920:
7919:
7884:
7880:
7874:
7870:
7858:
7847:
7826:
7823:
7822:
7788:
7784:
7769:
7765:
7750:
7746:
7741:
7738:
7737:
7734:
7717:
7711:
7703:
7683:
7654:
7651:
7650:
7629:
7625:
7616:
7612:
7610:
7607:
7606:
7583:
7579:
7555:
7551:
7540:
7537:
7536:
7502:
7499:
7498:
7469:
7465:
7441:
7437:
7426:
7423:
7422:
7406:
7403:
7402:
7386:
7383:
7382:
7366:
7358:
7350:
7347:
7346:
7300:
7297:
7296:
7279:
7275:
7273:
7270:
7269:
7253:
7250:
7249:
7233:
7230:
7229:
7209:
7206:
7205:
7129:
7126:
7125:
7103:
7100:
7099:
7077:
7074:
7073:
7047:
7044:
7043:
7027:
7024:
7023:
6990:
6986:
6978:
6975:
6974:
6951:
6947:
6939:
6936:
6935:
6919:
6916:
6915:
6898:
6894:
6892:
6889:
6888:
6872:
6869:
6868:
6849:
6846:
6845:
6829:
6826:
6825:
6799:
6795:
6794:
6789:
6781:
6778:
6777:
6760:
6756:
6729:
6725:
6723:
6720:
6719:
6693:
6689:
6676:
6668:
6665:
6664:
6641:
6637:
6631:
6627:
6612:
6607:
6596:
6583:
6582:
6581:
6572:
6555:
6541:
6531:
6522:
6518:
6516:
6513:
6512:
6489:
6485:
6479:
6475:
6460:
6455:
6444:
6431:
6430:
6429:
6420:
6403:
6389:
6374:
6370:
6369:
6357:
6353:
6351:
6342:
6338:
6336:
6333:
6332:
6297:
6294:
6293:
6277:
6274:
6273:
6254:
6220:
6203:
6200:
6199:
6177:
6174:
6173:
6163:Singleton bound
6119:
6115:
6113:
6110:
6109:
6084:
6026:
6022:
6004:
6000:
5982:
5978:
5969:
5965:
5933:
5930:
5929:
5904:
5901:
5900:
5884:
5882:
5879:
5878:
5853:
5850:
5849:
5817:
5813:
5804:
5800:
5768:
5765:
5764:
5739:
5736:
5735:
5719:
5716:
5715:
5690:
5687:
5686:
5669:
5665:
5648:
5645:
5644:
5627:
5623:
5614:
5610:
5589:
5585:
5570:
5566:
5564:
5561:
5560:
5538:
5535:
5534:
5500:
5496:
5487:
5483:
5450:
5446:
5444:
5441:
5440:
5414:
5410:
5408:
5405:
5404:
5388:
5385:
5384:
5359:
5356:
5355:
5327:
5324:
5323:
5306:
5302:
5300:
5297:
5296:
5271:
5268:
5267:
5239:
5236:
5235:
5210:
5207:
5206:
5181:
5178:
5177:
5152:
5149:
5148:
5123:
5120:
5119:
5094:
5091:
5090:
5065:
5062:
5061:
5045:
5042:
5041:
5016:
5013:
5012:
4960:
4957:
4956:
4949:
4914:
4910:
4895:
4891:
4882:
4878:
4873:
4867:
4863:
4857:
4853:
4847:
4836:
4810:
4809:
4797:
4793:
4778:
4774:
4765:
4761:
4760:
4756:
4755:
4751:
4743:
4741:
4738:
4737:
4713:
4710:
4709:
4684:
4680:
4659:
4655:
4637:
4633:
4615:
4611:
4602:
4598:
4566:
4562:
4555:
4551:
4531:
4527:
4520:
4516:
4505:
4501:
4494:
4490:
4473:
4470:
4469:
4454:
4451:
4450:
4425:
4422:
4421:
4399:
4396:
4395:
4370:
4367:
4366:
4341:
4338:
4337:
4312:
4309:
4308:
4283:
4280:
4279:
4254:
4251:
4250:
4222:
4219:
4218:
4196:
4193:
4192:
4167:
4164:
4163:
4138:
4135:
4134:
4131:
4093:
4089:
4074:
4070:
4040:
4036:
4021:
4017:
4015:
4012:
4011:
3995:
3991:
3982:
3978:
3976:
3973:
3972:
3936:
3935:
3920:
3916:
3907:
3903:
3900:
3899:
3893:
3892:
3883:
3879:
3870:
3866:
3863:
3862:
3853:
3849:
3840:
3836:
3829:
3828:
3811:
3808:
3807:
3790:
3786:
3784:
3781:
3780:
3773:
3751:
3747:
3745:
3742:
3741:
3718:
3714:
3699:
3695:
3693:
3690:
3689:
3663:
3659:
3650:
3646:
3628:
3624:
3615:
3611:
3609:
3606:
3605:
3589:
3586:
3585:
3561:
3560:
3545:
3541:
3532:
3528:
3525:
3524:
3518:
3517:
3508:
3504:
3495:
3491:
3488:
3487:
3478:
3474:
3465:
3461:
3454:
3453:
3436:
3433:
3432:
3407:
3403:
3388:
3384:
3382:
3379:
3378:
3358:
3355:
3354:
3337:
3333:
3331:
3328:
3327:
3275:
3269:
3265:
3253:
3249:
3240:
3236:
3234:
3231:
3230:
3214:
3211:
3210:
3193:
3189:
3187:
3184:
3183:
3166:
3162:
3160:
3157:
3156:
3149:
3128:
3125:
3124:
3100:
3097:
3096:
3072:
3071:
3059:
3055:
3052:
3051:
3045:
3044:
3038:
3034:
3031:
3030:
3024:
3020:
3013:
3012:
3005:
3004:
2992:
2981:
2975:
2970:
2964:
2953:
2947:
2935:
2931:
2929:
2923:
2922:
2917:
2912:
2907:
2902:
2896:
2895:
2883:
2878:
2872:
2867:
2861:
2856:
2850:
2844:
2840:
2838:
2832:
2831:
2819:
2814:
2808:
2803:
2797:
2792:
2786:
2780:
2776:
2774:
2764:
2763:
2737:
2734:
2733:
2717:
2714:
2713:
2697:
2694:
2693:
2671:
2668:
2667:
2633:
2630:
2629:
2609:
2606:
2605:
2587:
2586:
2571:
2567:
2558:
2554:
2551:
2550:
2544:
2543:
2534:
2530:
2521:
2517:
2514:
2513:
2504:
2500:
2491:
2487:
2480:
2479:
2462:
2459:
2458:
2441:
2437:
2428:
2424:
2416:
2413:
2412:
2396:
2393:
2392:
2369:
2365:
2350:
2346:
2344:
2341:
2340:
2324:
2321:
2320:
2303:
2299:
2297:
2294:
2293:
2277:
2274:
2273:
2252:
2248:
2242:
2238:
2226:
2215:
2193:
2189:
2187:
2184:
2183:
2166:
2162:
2160:
2157:
2156:
2139:
2135:
2117:
2113:
2098:
2094:
2083:
2080:
2079:
2072:
2064:systematic code
2038:
2034:
2019:
2015:
2013:
2010:
2009:
1968:
1964:
1962:
1959:
1958:
1928:
1905:
1902:
1901:
1900:code satisfies
1894:Singleton bound
1874:
1863:
1860:
1859:
1828:
1802:
1788:
1768:
1757:
1754:
1753:
1719:
1716:
1715:
1689:
1686:
1685:
1633:
1630:
1629:
1607:
1604:
1603:
1587:
1584:
1583:
1558:
1557:
1545:
1537:
1527:
1526:
1519:
1518:
1509:
1505:
1481:
1477:
1459:
1455:
1443:
1442:
1435:
1434:
1426:
1424:
1421:
1420:
1404:
1402:
1399:
1398:
1382:}, ... , where
1326:
1322:
1307:
1303:
1301:
1298:
1297:
1269:
1266:
1265:
1249:
1246:
1245:
1225:
1222:
1221:
1205:
1202:
1201:
1194:
1161:
1158:
1157:
1135:
1132:
1131:
1110:
1102:
1099:
1098:
1075:
1072:
1071:
1055:
1052:
1051:
1035:
1032:
1031:
1000:
997:
996:
977:
974:
973:
954:
951:
950:
931:
928:
927:
920:
814:Pioneer, Venus
745:Mars Pathfinder
721:Voyager program
705:
627:
592:
536:
531:
509:was developed.
483:(DVB) standard
469:digital storage
455:in the form of
453:Voyager program
424:Elwyn Berlekamp
376:Gustave Solomon
368:
352:
348:
341:
333:
321:
313:
309:
305:
301:
239:Gustave Solomon
229:are a group of
220:
176:
172:
158:
135:
67:
65:Polynomial code
63:
46:Gustave Solomon
28:
23:
22:
15:
12:
11:
5:
22582:
22572:
22571:
22566:
22552:
22551:
22546:
22541:
22536:
22531:
22526:
22521:
22514:
22511:
22510:
22509:
22496:
22486:
22471:
22466:
22461:
22456:
22451:
22443:
22440:
22438:
22437:External links
22435:
22434:
22433:
22428:
22410:
22405:
22386:
22371:
22362:(4): 549–557,
22346:
22326:
22321:
22305:
22285:
22271:
22262:(2): 300–304,
22244:
22235:(4): 459–470,
22221:
22185:
22180:
22167:
22158:(8): 2324–33,
22140:
22110:
22107:
22104:
22103:
22067:
22053:10.1.1.13.2021
22028:
21996:
21989:
21949:
21942:
21902:
21889:10.1.1.115.292
21864:
21855:
21843:
21834:
21813:(8): 873–883,
21790:
21765:
21752:
21707:
21692:
21674:
21668:
21643:
21620:
21587:
21572:
21552:
21529:
21510:(2): 207–214.
21494:
21482:
21481:
21479:
21476:
21473:
21472:
21468:bit error rate
21458:
21457:
21455:
21452:
21451:
21450:
21445:
21440:
21435:
21430:
21425:
21420:
21413:
21410:
21402:
21335:
21315:
21287:
21246:
21233:
21211:
21198:
21195:
21194:
21191:
21188:
21184:
21183:
21180:
21177:
21173:
21172:
21169:
21166:
21162:
21161:
21158:
21155:
21151:
21150:
21145:
21142:
21137:
21134:
21126:
21125:
21114:
21111:
21106:
21102:
21091:
21080:
21077:
21072:
21069:
21065:
21054:
21035:
21032:
21027:
21023:
21019:
21016:
21013:
21008:
21004:
21000:
20980:
20975:
20971:
20960:
20949:
20944:
20940:
20936:
20933:
20930:
20925:
20920:
20917:
20914:
20910:
20906:
20901:
20898:
20894:
20878:
20875:
20870:
20867:
20859:
20806:
20786:
20758:
20744:
20738:
20735:
20734:
20731:
20728:
20727:
20724:
20721:
20720:
20717:
20714:
20713:
20710:
20707:
20706:
20703:
20700:
20699:
20696:
20693:
20692:
20690:
20685:
20680:
20672:
20668:
20664:
20663:
20658:
20654:
20650:
20649:
20644:
20640:
20636:
20635:
20630:
20626:
20622:
20621:
20616:
20612:
20608:
20607:
20602:
20598:
20594:
20593:
20588:
20584:
20580:
20579:
20577:
20570:
20564:
20561:
20559:
20556:
20554:
20551:
20549:
20546:
20544:
20541:
20539:
20536:
20534:
20531:
20530:
20527:
20524:
20522:
20519:
20517:
20514:
20512:
20509:
20507:
20504:
20502:
20499:
20497:
20494:
20493:
20490:
20487:
20485:
20482:
20480:
20477:
20475:
20472:
20470:
20467:
20465:
20462:
20460:
20457:
20456:
20453:
20450:
20448:
20445:
20443:
20440:
20438:
20435:
20433:
20430:
20428:
20425:
20423:
20420:
20419:
20416:
20413:
20411:
20408:
20406:
20403:
20401:
20398:
20396:
20393:
20391:
20388:
20386:
20383:
20382:
20379:
20376:
20374:
20371:
20369:
20366:
20364:
20361:
20359:
20356:
20354:
20351:
20349:
20346:
20345:
20342:
20339:
20337:
20334:
20332:
20329:
20327:
20324:
20322:
20319:
20317:
20314:
20312:
20309:
20308:
20306:
20274:
20268:
20265:
20264:
20261:
20258:
20257:
20254:
20251:
20250:
20247:
20244:
20243:
20240:
20237:
20236:
20233:
20230:
20229:
20226:
20223:
20222:
20220:
20215:
20210:
20202:
20198:
20194:
20193:
20188:
20184:
20180:
20179:
20174:
20170:
20166:
20165:
20160:
20156:
20152:
20151:
20146:
20142:
20138:
20137:
20132:
20128:
20124:
20123:
20118:
20114:
20110:
20109:
20107:
20100:
20094:
20091:
20089:
20086:
20084:
20081:
20079:
20076:
20074:
20071:
20069:
20066:
20064:
20061:
20060:
20057:
20054:
20052:
20049:
20047:
20044:
20042:
20039:
20037:
20034:
20032:
20029:
20027:
20024:
20023:
20020:
20017:
20015:
20012:
20010:
20007:
20005:
20002:
20000:
19997:
19995:
19992:
19990:
19987:
19986:
19983:
19980:
19978:
19975:
19973:
19970:
19968:
19965:
19963:
19960:
19958:
19955:
19953:
19950:
19949:
19946:
19943:
19941:
19938:
19936:
19933:
19931:
19928:
19926:
19923:
19921:
19918:
19916:
19913:
19912:
19909:
19906:
19904:
19901:
19899:
19896:
19894:
19891:
19889:
19886:
19884:
19881:
19879:
19876:
19875:
19872:
19869:
19867:
19864:
19862:
19859:
19857:
19854:
19852:
19849:
19847:
19844:
19842:
19839:
19838:
19836:
19811:
19806:
19802:
19796:
19792:
19788:
19785:
19782:
19777:
19772:
19768:
19762:
19758:
19754:
19749:
19744:
19740:
19734:
19730:
19726:
19721:
19716:
19712:
19706:
19702:
19698:
19693:
19689:
19683:
19679:
19675:
19670:
19666:
19662:
19659:
19656:
19651:
19647:
19641:
19637:
19633:
19628:
19624:
19620:
19615:
19611:
19582:
19579:
19576:
19571:
19567:
19563:
19560:
19557:
19554:
19549:
19545:
19541:
19538:
19533:
19529:
19500:
19489:
19466:
19455:
19444:
19437:
19434:
19421:
19401:
19396:
19392:
19388:
19385:
19368:
19365:
19352:
19349:
19346:
19343:
19340:
19317:
19314:
19311:
19308:
19305:
19302:
19299:
19296:
19291:
19288:
19282:
19276:
19271:
19268:
19263:
19233:
19229:
19206:
19202:
19186:
19183:
19177:
19174:
17969:% polynomials)
17648:
17642:
17639:
17347:
17337:
17334:
17332:
17331:MATLAB example
17329:
17298:
17295:
17265:
17260:
17256:
17252:
17249:
17246:
17203:−1) / 2⌋
17163:
17160:
17101:
17098:
17095:
17092:
17089:
17081:
17079:
17074:
17071:
17068:
17064:
17058:
17054:
17050:
17047:
17044:
17039:
17036:
17033:
17029:
17023:
17019:
17015:
17010:
17007:
17004:
17000:
16994:
16990:
16986:
16983:
16980:
16977:
16975:
16971:
16967:
16963:
16962:
16959:
16954:
16950:
16944:
16941:
16938:
16934:
16930:
16927:
16924:
16919:
16916:
16913:
16909:
16903:
16899:
16895:
16890:
16886:
16882:
16875:
16871:
16867:
16862:
16859:
16856:
16854:
16850:
16846:
16842:
16841:
16819:
16815:
16792:
16788:
16750:
16746:
16723:
16719:
16696:
16693:
16690:
16687:
16684:
16675:
16670:
16666:
16662:
16659:
16656:
16651:
16647:
16643:
16638:
16634:
16630:
16625:
16621:
16441:
16438:
16429:
16420:
16407:
16398:
16395:
16394:
16383:
16376:
16372:
16371:
16364:
16353:
16349:
16348:
16345:
16330:
16326:
16325:
16322:
16303:
16299:
16298:
16293:
16290:
16285:
16282:
16270:
16267:
16262:
16256:
16245:
16236:
16227:
16220:
16211:
16202:
16195:
16171:
16158:
16149:
16140:
16127:
16118:
16109:
16099:
16074:
16059:
16052:
16037:
16027:
16023:
15990:
15986:
15971:
15967:
15952:
15948:
15933:
15909:
15892:
15871:
15867:
15847:
15839:
15835:
15831:
15830:
15827:
15822:
15818:
15814:
15813:
15808:
15804:
15798:
15794:
15790:
15789:
15786:
15783:
15782:
15779:
15776:
15775:
15772:
15769:
15768:
15763:
15759:
15753:
15749:
15745:
15744:
15739:
15735:
15729:
15725:
15721:
15720:
15715:
15711:
15705:
15701:
15697:
15696:
15694:
15689:
15684:
15676:
15672:
15668:
15667:
15664:
15661:
15657:
15653:
15647:
15643:
15639:
15634:
15630:
15626:
15625:
15620:
15616:
15612:
15608:
15604:
15598:
15594:
15590:
15585:
15581:
15575:
15571:
15567:
15562:
15558:
15554:
15553:
15548:
15544:
15540:
15536:
15532:
15526:
15522:
15518:
15513:
15509:
15503:
15499:
15495:
15490:
15486:
15480:
15476:
15472:
15467:
15463:
15459:
15458:
15453:
15449:
15445:
15441:
15437:
15431:
15427:
15423:
15418:
15414:
15408:
15404:
15400:
15395:
15391:
15385:
15381:
15377:
15372:
15368:
15364:
15363:
15358:
15354:
15350:
15346:
15342:
15336:
15332:
15328:
15323:
15319:
15313:
15309:
15305:
15300:
15296:
15290:
15286:
15282:
15277:
15273:
15269:
15268:
15263:
15259:
15255:
15251:
15247:
15241:
15237:
15233:
15228:
15224:
15218:
15214:
15210:
15205:
15201:
15195:
15191:
15187:
15186:
15181:
15177:
15173:
15169:
15165:
15159:
15155:
15151:
15146:
15142:
15136:
15132:
15128:
15127:
15122:
15118:
15114:
15110:
15106:
15100:
15096:
15092:
15091:
15089:
15057:
15054:
15051:
15048:
15045:
15040:
15036:
15032:
15029:
15026:
15023:
15020:
15017:
15014:
15011:
15008:
15005:
15002:
14999:
14996:
14968:
14964:
14960:
14957:
14952:
14948:
14944:
14941:
14938:
14933:
14930:
14927:
14923:
14917:
14914:
14911:
14907:
14903:
14898:
14894:
14888:
14884:
14880:
14877:
14875:
14873:
14870:
14867:
14864:
14861:
14860:
14857:
14854:
14851:
14846:
14842:
14838:
14835:
14832:
14827:
14824:
14821:
14817:
14811:
14808:
14805:
14801:
14797:
14792:
14788:
14782:
14778:
14774:
14771:
14769:
14767:
14764:
14761:
14758:
14755:
14754:
14749:
14745:
14741:
14738:
14733:
14729:
14725:
14722:
14719:
14714:
14711:
14708:
14704:
14698:
14695:
14692:
14688:
14684:
14679:
14676:
14673:
14669:
14663:
14659:
14655:
14652:
14650:
14648:
14645:
14642:
14639:
14636:
14635:
14622:syndromes and
14592:
14589:
14575:
14574:
14571:
14568:
14561:
14550:
14547:
14544:
14540:
14539:
14536:
14533:
14526:
14515:
14512:
14509:
14505:
14504:
14501:
14498:
14495:
14488:
14485:
14482:
14478:
14477:
14474:
14471:
14468:
14461:
14458:
14455:
14451:
14450:
14445:
14440:
14435:
14430:
14425:
14419:
14414:
14402:
14399:
14360:
14357:
14354:
14350:
14341:
14337:
14333:
14330:
14327:
14322:
14319:
14316:
14312:
14303:
14299:
14295:
14290:
14286:
14282:
14279:
14255:
14252:
14225:
14221:
14217:
14214:
14209:
14205:
14201:
14198:
14193:
14189:
14183:
14179:
14175:
14170:
14166:
14160:
14156:
14140:
14136:
14132:
14130:
14125:
14121:
14117:
14115:
14112:
14109:
14096:
14088:
14084:= 757 = 3 and
14081:
14076:
14062:
14056:
14053:
14052:
14049:
14046:
14045:
14043:
14038:
14033:
14025:
14021:
14017:
14016:
14011:
14007:
14003:
14002:
14000:
13993:
13987:
13984:
13982:
13979:
13978:
13975:
13972:
13970:
13967:
13966:
13964:
13934:
13928:
13925:
13924:
13921:
13918:
13917:
13915:
13910:
13905:
13899:
13896:
13893:
13892:
13889:
13886:
13883:
13882:
13880:
13875:
13870:
13862:
13858:
13854:
13853:
13848:
13844:
13840:
13839:
13837:
13830:
13824:
13821:
13819:
13816:
13815:
13812:
13809:
13807:
13804:
13803:
13801:
13780:
13777:
13774:
13769:
13765:
13761:
13758:
13755:
13750:
13746:
13741:
13738:
13735:
13732:
13727:
13723:
13719:
13716:
13713:
13708:
13704:
13699:
13696:
13693:
13690:
13685:
13681:
13677:
13674:
13671:
13666:
13662:
13642:
13639:
13636:
13633:
13630:
13627:
13624:
13621:
13616:
13612:
13608:
13605:
13602:
13597:
13593:
13589:
13586:
13583:
13578:
13574:
13570:
13567:
13564:
13559:
13555:
13551:
13548:
13545:
13540:
13536:
13532:
13529:
13526:
13523:
13518:
13514:
13510:
13507:
13504:
13499:
13495:
13466:
13463:
13460:
13457:
13454:
13449:
13445:
13441:
13438:
13433:
13429:
13425:
13422:
13417:
13413:
13409:
13406:
13401:
13397:
13393:
13390:
13385:
13381:
13377:
13374:
13371:
13368:
13365:
13362:
13359:
13356:
13353:
13350:
13347:
13344:
13341:
13338:
13335:
13332:
13312:
13309:
13306:
13303:
13300:
13295:
13291:
13287:
13284:
13279:
13275:
13271:
13268:
13263:
13259:
13255:
13252:
13247:
13243:
13239:
13236:
13231:
13227:
13223:
13220:
13217:
13214:
13211:
13206:
13202:
13198:
13193:
13189:
13184:
13181:
13178:
13175:
13172:
13169:
13166:
13163:
13160:
13141:
13138:
13135:
13132:
13129:
13124:
13120:
13116:
13113:
13108:
13104:
13100:
13097:
13094:
13091:
13088:
13083:
13079:
13073:
13069:
13064:
13061:
13058:
13055:
13052:
13049:
13046:
13043:
13038:
13034:
12994:
12991:
12988:
12985:
12982:
12977:
12973:
12969:
12966:
12961:
12957:
12953:
12950:
12945:
12941:
12937:
12934:
12929:
12925:
12921:
12918:
12915:
12912:
12907:
12903:
12899:
12896:
12893:
12890:
12885:
12881:
12877:
12874:
12871:
12868:
12865:
12862:
12859:
12856:
12853:
12850:
12847:
12844:
12841:
12804:
12801:
12778:
12776:
12769:
12755:
12754:Fix the errors
12752:
12746:
12732:
12719:
12713:
12710:
12696:
12689:
12683:
12680:
12661:
12658:
12653:
12649:
12626:
12622:
12608:
12600:
12596:
12593:
12591:, p. 35)
12557:
12549:
12546:
12543:
12539:
12535:
12532:
12531:
12528:
12525:
12524:
12519:
12516:
12513:
12509:
12505:
12502:
12501:
12496:
12493:
12490:
12486:
12482:
12479:
12478:
12476:
12471:
12466:
12458:
12454:
12450:
12449:
12446:
12443:
12442:
12437:
12434:
12431:
12427:
12423:
12422:
12417:
12413:
12409:
12408:
12406:
12399:
12391:
12388:
12385:
12382:
12378:
12374:
12372:
12369:
12365:
12362:
12359:
12355:
12351:
12347:
12343:
12339:
12338:
12335:
12332:
12330:
12327:
12325:
12322:
12320:
12317:
12316:
12311:
12308:
12305:
12301:
12297:
12295:
12292:
12288:
12284:
12280:
12276:
12272:
12268:
12267:
12262:
12258:
12254:
12252:
12249:
12245:
12241:
12237:
12233:
12229:
12225:
12224:
12222:
12206:
12178:
12175:
12172:
12168:
12164:
12161:
12156:
12152:
12146:
12143:
12140:
12137:
12134:
12130:
12126:
12123:
12120:
12115:
12112:
12109:
12105:
12099:
12096:
12093:
12089:
12085:
12080:
12076:
12070:
12066:
12043:
12040:
12037:
12033:
12010:
12007:
12002:
11998:
11992:
11988:
11984:
11979:
11976:
11973:
11969:
11963:
11960:
11957:
11953:
11949:
11946:
11943:
11938:
11935:
11932:
11929:
11926:
11922:
11916:
11912:
11908:
11903:
11900:
11897:
11893:
11870:
11867:
11863:
11857:
11852:
11848:
11842:
11838:
11832:
11827:
11824:
11821:
11817:
11812:
11806:
11802:
11798:
11795:
11792:
11788:
11782:
11779:
11776:
11773:
11770:
11765:
11761:
11755:
11751:
11745:
11740:
11737:
11734:
11730:
11725:
11719:
11715:
11711:
11707:
11701:
11698:
11695:
11692:
11689:
11684:
11680:
11674:
11670:
11664:
11659:
11656:
11653:
11649:
11644:
11638:
11634:
11630:
11626:
11620:
11617:
11614:
11609:
11605:
11599:
11595:
11589:
11584:
11581:
11578:
11574:
11569:
11548:
11526:
11523:
11519:
11513:
11508:
11504:
11498:
11494:
11488:
11484:
11478:
11473:
11470:
11467:
11463:
11458:
11454:
11451:
11448:
11444:
11438:
11435:
11432:
11429:
11426:
11421:
11417:
11411:
11407:
11401:
11397:
11391:
11386:
11383:
11380:
11376:
11371:
11367:
11363:
11357:
11354:
11351:
11348:
11345:
11340:
11336:
11330:
11326:
11320:
11316:
11310:
11305:
11302:
11299:
11295:
11290:
11286:
11282:
11276:
11273:
11270:
11265:
11261:
11255:
11251:
11245:
11240:
11237:
11234:
11230:
11225:
11202:
11199:
11195:
11189:
11184:
11180:
11174:
11170:
11164:
11160:
11156:
11153:
11150:
11145:
11142:
11139:
11136:
11133:
11128:
11124:
11118:
11114:
11108:
11104:
11100:
11095:
11092:
11089:
11086:
11083:
11078:
11074:
11068:
11064:
11058:
11054:
11050:
11045:
11042:
11039:
11034:
11030:
11024:
11020:
11015:
11009:
11004:
11001:
10998:
10994:
10959:
10956:
10951:
10946:
10942:
10936:
10932:
10926:
10922:
10918:
10915:
10912:
10907:
10904:
10901:
10898:
10895:
10890:
10886:
10880:
10876:
10870:
10866:
10862:
10857:
10854:
10851:
10848:
10845:
10840:
10836:
10830:
10826:
10820:
10816:
10812:
10807:
10804:
10801:
10796:
10792:
10786:
10782:
10778:
10776:
10773:
10770:
10765:
10762:
10757:
10753:
10747:
10744:
10741:
10736:
10732:
10726:
10722:
10716:
10712:
10708:
10705:
10702:
10697:
10694:
10689:
10685:
10679:
10676:
10673:
10668:
10664:
10658:
10654:
10648:
10644:
10640:
10635:
10632:
10627:
10623:
10617:
10614:
10611:
10606:
10602:
10596:
10592:
10586:
10582:
10578:
10573:
10570:
10567:
10562:
10558:
10552:
10548:
10544:
10542:
10539:
10536:
10532:
10526:
10523:
10518:
10514:
10508:
10504:
10500:
10497:
10494:
10489:
10486:
10481:
10477:
10471:
10467:
10463:
10458:
10455:
10450:
10446:
10440:
10436:
10432:
10429:
10425:
10419:
10416:
10413:
10408:
10404:
10398:
10394:
10390:
10388:
10385:
10382:
10379:
10374:
10371:
10366:
10362:
10358:
10355:
10350:
10347:
10344:
10339:
10335:
10329:
10325:
10321:
10319:
10297:
10294:
10291:
10286:
10282:
10276:
10272:
10251:
10248:
10245:
10242:
10239:
10219:
10197:
10194:
10191:
10186:
10183:
10178:
10174:
10170:
10167:
10147:
10144:
10141:
10138:
10135:
10132:
10129:
10124:
10120:
10116:
10111:
10108:
10103:
10099:
10095:
10092:
10089:
10067:
10064:
10059:
10055:
10051:
10048:
10026:
10023:
10018:
10014:
9981:
9977:
9971:
9967:
9963:
9960:
9957:
9952:
9948:
9942:
9938:
9934:
9929:
9925:
9919:
9915:
9911:
9908:
9905:
9902:
9897:
9893:
9889:
9886:
9883:
9880:
9875:
9870:
9867:
9864:
9860:
9856:
9853:
9850:
9847:
9844:
9817:
9811:
9808:
9795:
9775:
9772:
9749:
9746:
9743:
9730:
9717:
9711:
9693:
9685:
9682:
9679:
9675:
9671:
9670:
9667:
9664:
9663:
9658:
9654:
9650:
9649:
9644:
9640:
9636:
9635:
9633:
9628:
9623:
9615:
9611:
9607:
9606:
9603:
9600:
9599:
9594:
9590:
9586:
9585:
9580:
9576:
9572:
9571:
9569:
9562:
9554:
9551:
9548:
9543:
9539:
9535:
9533:
9530:
9526:
9523:
9520:
9515:
9511:
9507:
9503:
9500:
9497:
9492:
9488:
9484:
9483:
9480:
9477:
9475:
9472:
9470:
9467:
9465:
9462:
9461:
9456:
9451:
9447:
9443:
9441:
9438:
9434:
9429:
9425:
9421:
9417:
9412:
9408:
9404:
9403:
9398:
9393:
9389:
9385:
9383:
9380:
9376:
9371:
9367:
9363:
9359:
9354:
9350:
9346:
9345:
9343:
9332:error values.
9328:
9321:
9314:
9307:equations in 2
9277:
9272:
9268:
9264:
9259:
9255:
9247:
9243:
9238:
9234:
9231:
9224:
9220:
9216:
9213:
9209:
9205:
9198:
9194:
9189:
9183:
9179:
9175:
9151:
9146:
9142:
9136:
9132:
9126:
9121:
9118:
9115:
9111:
9107:
9102:
9098:
9071:
9067:
9062:
9058:
9053:
9049:
9042:
9035:
9031:
9026:
9022:
9017:
9013:
8999:
8989:
8985:error locators
8980:
8977:
8958:
8954:
8933:
8930:
8927:
8924:
8904:
8901:
8898:
8893:
8889:
8885:
8882:
8858:
8855:
8852:
8849:
8846:
8843:
8840:
8837:
8834:
8831:
8828:
8824:
8817:
8813:
8807:
8802:
8798:
8794:
8785:
8781:
8776:
8770:
8765:
8762:
8759:
8755:
8751:
8748:
8746:
8744:
8741:
8736:
8732:
8728:
8725:
8722:
8719:
8717:
8715:
8712:
8707:
8703:
8699:
8696:
8693:
8690:
8687:
8684:
8679:
8675:
8671:
8668:
8665:
8662:
8657:
8653:
8649:
8646:
8643:
8640:
8635:
8631:
8627:
8624:
8621:
8618:
8616:
8612:
8608:
8604:
8603:
8589:
8572:
8569:
8566:
8562:
8558:
8553:
8549:
8536:
8533:
8502:
8500:
8493:
8469:
8465:
8460:
8452:
8448:
8443:
8437:
8432:
8429:
8426:
8422:
8418:
8415:
8412:
8409:
8406:
8389:
8374:
8356:
8352:
8346:
8342:
8336:
8333:
8330:
8325:
8322:
8319:
8315:
8311:
8308:
8305:
8302:
8299:
8280:
8277:
8274:
8271:
8268:
8265:
8262:
8259:
8256:
8253:
8250:
8247:
8244:
8241:
8203:
8200:
8197:
8194:
8191:
8188:
8185:
8182:
8179:
8176:
8173:
8167:
8164:
8161:
8158:
8153:
8149:
8145:
8142:
8122:
8119:
8116:
8111:
8107:
8103:
8100:
8095:
8091:
8087:
8084:
8081:
8078:
8075:
8072:
8067:
8063:
8059:
8056:
8051:
8047:
8043:
8040:
8037:
8034:
7991:
7988:
7983:
7979:
7975:
7972:
7969:
7964:
7961:
7958:
7953:
7950:
7947:
7943:
7939:
7936:
7933:
7930:
7927:
7887:
7883:
7877:
7873:
7867:
7864:
7861:
7856:
7853:
7850:
7846:
7842:
7839:
7836:
7833:
7830:
7802:
7797:
7794:
7791:
7787:
7783:
7780:
7777:
7772:
7768:
7764:
7761:
7758:
7753:
7749:
7745:
7733:
7730:
7713:Main article:
7710:
7707:
7702:
7699:
7682:
7679:
7658:
7632:
7628:
7624:
7619:
7615:
7594:
7591:
7586:
7582:
7578:
7575:
7572:
7569:
7566:
7563:
7558:
7554:
7550:
7547:
7544:
7524:
7521:
7518:
7515:
7512:
7509:
7506:
7486:
7483:
7478:
7475:
7472:
7468:
7464:
7461:
7458:
7455:
7452:
7449:
7444:
7440:
7436:
7433:
7430:
7410:
7390:
7369:
7365:
7361:
7357:
7354:
7334:
7331:
7328:
7325:
7322:
7319:
7316:
7313:
7310:
7307:
7304:
7282:
7278:
7268:take the form
7257:
7237:
7226:primitive root
7213:
7163:
7160:
7157:
7154:
7151:
7148:
7145:
7142:
7139:
7136:
7133:
7113:
7110:
7107:
7087:
7084:
7081:
7057:
7054:
7051:
7031:
7007:
7004:
7001:
6998:
6993:
6989:
6985:
6982:
6962:
6959:
6954:
6950:
6946:
6943:
6923:
6901:
6897:
6876:
6853:
6833:
6810:
6807:
6802:
6798:
6793:
6788:
6785:
6763:
6759:
6755:
6752:
6749:
6746:
6743:
6740:
6737:
6732:
6728:
6707:
6704:
6701:
6696:
6692:
6688:
6683:
6680:
6675:
6672:
6650:
6647:
6644:
6640:
6634:
6630:
6626:
6623:
6620:
6615:
6610:
6606:
6599:
6594:
6591:
6586:
6580:
6575:
6570:
6567:
6564:
6561:
6558:
6554:
6548:
6545:
6538:
6535:
6530:
6525:
6521:
6498:
6495:
6492:
6488:
6482:
6478:
6474:
6471:
6468:
6463:
6458:
6454:
6447:
6442:
6439:
6434:
6428:
6423:
6418:
6415:
6412:
6409:
6406:
6402:
6396:
6393:
6385:
6382:
6377:
6373:
6366:
6363:
6360:
6356:
6350:
6345:
6341:
6301:
6281:
6227:
6223:
6219:
6216:
6213:
6210:
6207:
6187:
6184:
6181:
6142:
6139:
6136:
6133:
6130:
6127:
6122:
6118:
6083:
6080:
6068:
6064:
6061:
6058:
6055:
6050:
6046:
6043:
6040:
6037:
6034:
6029:
6025:
6021:
6018:
6015:
6012:
6007:
6003:
5999:
5996:
5993:
5990:
5985:
5981:
5977:
5972:
5968:
5964:
5961:
5958:
5955:
5952:
5949:
5946:
5943:
5940:
5937:
5917:
5914:
5911:
5908:
5887:
5866:
5863:
5860:
5857:
5835:
5831:
5828:
5825:
5820:
5816:
5812:
5807:
5803:
5799:
5796:
5793:
5790:
5787:
5784:
5781:
5778:
5775:
5772:
5752:
5749:
5746:
5743:
5723:
5703:
5700:
5697:
5694:
5672:
5668:
5664:
5661:
5658:
5655:
5652:
5630:
5626:
5622:
5617:
5613:
5609:
5606:
5603:
5598:
5595:
5592:
5588:
5584:
5579:
5576:
5573:
5569:
5548:
5545:
5542:
5520:
5517:
5514:
5511:
5508:
5499:
5490:
5486:
5482:
5479:
5476:
5473:
5470:
5467:
5464:
5461:
5458:
5453:
5449:
5428:
5425:
5422:
5417:
5413:
5392:
5372:
5369:
5366:
5363:
5343:
5340:
5337:
5334:
5331:
5309:
5305:
5284:
5281:
5278:
5275:
5252:
5249:
5246:
5243:
5223:
5220:
5217:
5214:
5194:
5191:
5188:
5185:
5165:
5162:
5159:
5156:
5136:
5133:
5130:
5127:
5107:
5104:
5101:
5098:
5078:
5075:
5072:
5069:
5049:
5029:
5026:
5023:
5020:
5000:
4997:
4994:
4991:
4988:
4985:
4982:
4979:
4976:
4973:
4970:
4967:
4964:
4948:
4945:
4933:
4929:
4923:
4920:
4917:
4913:
4909:
4906:
4903:
4898:
4894:
4890:
4885:
4881:
4870:
4866:
4860:
4856:
4850:
4845:
4842:
4839:
4835:
4831:
4828:
4825:
4822:
4819:
4813:
4806:
4800:
4796:
4792:
4789:
4786:
4781:
4777:
4773:
4768:
4764:
4759:
4754:
4750:
4746:
4723:
4720:
4717:
4693:
4690:
4687:
4683:
4679:
4674:
4671:
4668:
4665:
4662:
4658:
4652:
4649:
4646:
4643:
4640:
4636:
4632:
4629:
4626:
4623:
4618:
4614:
4610:
4605:
4601:
4597:
4593:
4587:
4584:
4581:
4578:
4575:
4572:
4569:
4565:
4561:
4558:
4554:
4550:
4546:
4540:
4537:
4534:
4530:
4526:
4523:
4519:
4514:
4508:
4504:
4500:
4497:
4493:
4489:
4486:
4483:
4480:
4477:
4458:
4438:
4435:
4432:
4429:
4409:
4406:
4403:
4383:
4380:
4377:
4374:
4351:
4348:
4345:
4325:
4322:
4319:
4316:
4296:
4293:
4290:
4287:
4267:
4264:
4261:
4258:
4238:
4235:
4232:
4229:
4226:
4206:
4203:
4200:
4180:
4177:
4174:
4171:
4151:
4148:
4145:
4142:
4130:
4127:
4107:
4102:
4099:
4096:
4092:
4088:
4085:
4082:
4077:
4073:
4069:
4066:
4063:
4060:
4057:
4054:
4049:
4046:
4043:
4039:
4035:
4032:
4029:
4024:
4020:
3998:
3994:
3990:
3985:
3981:
3940:
3934:
3929:
3926:
3923:
3919:
3915:
3910:
3906:
3902:
3901:
3898:
3895:
3894:
3891:
3886:
3882:
3878:
3873:
3869:
3865:
3864:
3861:
3856:
3852:
3848:
3843:
3839:
3835:
3834:
3832:
3827:
3824:
3821:
3818:
3815:
3793:
3789:
3772:
3769:
3754:
3750:
3727:
3724:
3721:
3717:
3713:
3710:
3707:
3702:
3698:
3688:, are exactly
3677:
3672:
3669:
3666:
3662:
3658:
3653:
3649:
3645:
3642:
3639:
3636:
3631:
3627:
3623:
3618:
3614:
3593:
3565:
3559:
3554:
3551:
3548:
3544:
3540:
3535:
3531:
3527:
3526:
3523:
3520:
3519:
3516:
3511:
3507:
3503:
3498:
3494:
3490:
3489:
3486:
3481:
3477:
3473:
3468:
3464:
3460:
3459:
3457:
3452:
3449:
3446:
3443:
3440:
3416:
3413:
3410:
3406:
3402:
3399:
3396:
3391:
3387:
3373:, one can use
3362:
3340:
3336:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3291:
3288:
3285:
3282:
3272:
3268:
3264:
3261:
3256:
3252:
3248:
3243:
3239:
3218:
3196:
3192:
3169:
3165:
3148:
3145:
3132:
3104:
3076:
3068:
3065:
3062:
3058:
3054:
3053:
3050:
3047:
3046:
3041:
3037:
3033:
3032:
3027:
3023:
3019:
3018:
3016:
3009:
3001:
2998:
2995:
2990:
2987:
2984:
2980:
2976:
2974:
2971:
2967:
2962:
2959:
2956:
2952:
2948:
2944:
2941:
2938:
2934:
2930:
2928:
2925:
2924:
2921:
2918:
2916:
2913:
2911:
2908:
2906:
2903:
2901:
2898:
2897:
2892:
2889:
2886:
2881:
2877:
2873:
2871:
2868:
2864:
2859:
2855:
2851:
2847:
2843:
2839:
2837:
2834:
2833:
2828:
2825:
2822:
2817:
2813:
2809:
2807:
2804:
2800:
2795:
2791:
2787:
2783:
2779:
2775:
2773:
2770:
2769:
2767:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2721:
2701:
2681:
2678:
2675:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2626:linear mapping
2613:
2604:This function
2591:
2585:
2580:
2577:
2574:
2570:
2566:
2561:
2557:
2553:
2552:
2549:
2546:
2545:
2542:
2537:
2533:
2529:
2524:
2520:
2516:
2515:
2512:
2507:
2503:
2499:
2494:
2490:
2486:
2485:
2483:
2478:
2475:
2472:
2469:
2466:
2444:
2440:
2436:
2431:
2427:
2423:
2420:
2400:
2378:
2375:
2372:
2368:
2364:
2361:
2358:
2353:
2349:
2328:
2306:
2302:
2281:
2261:
2255:
2251:
2245:
2241:
2235:
2232:
2229:
2224:
2221:
2218:
2214:
2210:
2207:
2204:
2201:
2196:
2192:
2169:
2165:
2142:
2138:
2134:
2131:
2126:
2123:
2120:
2116:
2112:
2109:
2106:
2101:
2097:
2093:
2090:
2087:
2078:, the message
2071:
2068:
2041:
2037:
2033:
2030:
2027:
2022:
2018:
1971:
1967:
1935:
1931:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1881:
1877:
1873:
1870:
1867:
1847:
1844:
1841:
1838:
1835:
1831:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1805:
1801:
1798:
1795:
1791:
1787:
1784:
1781:
1778:
1775:
1771:
1767:
1764:
1761:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1699:
1696:
1693:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1617:
1614:
1611:
1591:
1578:Since any two
1567:
1561:
1555:
1552:
1544:
1536:
1530:
1522:
1517:
1512:
1508:
1504:
1501:
1498:
1495:
1492:
1489:
1484:
1480:
1476:
1473:
1470:
1467:
1462:
1458:
1454:
1451:
1446:
1438:
1433:
1429:
1407:
1329:
1325:
1321:
1318:
1315:
1310:
1306:
1273:
1253:
1229:
1209:
1193:
1190:
1177:
1174:
1171:
1168:
1165:
1145:
1142:
1139:
1117:
1114:
1109:
1106:
1079:
1059:
1039:
1016:
1013:
1010:
1007:
1004:
981:
971:message length
958:
935:
919:
916:
913:
912:
909:
904:
900:
899:
896:
891:
887:
886:
883:
880:
876:
875:
872:
869:
865:
864:
861:
858:
854:
853:
850:
847:
843:
842:
839:
834:
830:
829:
826:
820:
816:
815:
812:
811:(CC) (25, 1/2)
806:
802:
801:
798:
795:
791:
790:
787:
784:
704:
701:
626:
623:
591:
588:
535:
532:
530:
527:
372:Irving S. Reed
367:
364:
345: + 1
235:Irving S. Reed
222:
221:
219:
218:
211:
204:
196:
193:
192:
186:
185:
181:
180:
166:
165:
161:
160:
154:
152:
146:
145:
117:
111:
110:
100:
94:
93:
88:
86:Message length
82:
81:
76:
70:
69:
58:
54:
53:
52:Classification
49:
48:
42:Irving S. Reed
39:
35:
34:
26:
9:
6:
4:
3:
2:
22581:
22570:
22569:Coding theory
22567:
22565:
22562:
22561:
22559:
22550:
22547:
22545:
22542:
22540:
22537:
22535:
22532:
22530:
22527:
22525:
22522:
22520:
22517:
22516:
22504:
22503:
22497:
22494:
22490:
22487:
22483:
22479:
22478:
22472:
22470:
22467:
22465:
22462:
22460:
22457:
22455:
22452:
22449:
22446:
22445:
22431:
22429:9781461550051
22425:
22421:
22420:
22415:
22411:
22408:
22402:
22398:
22397:
22392:
22387:
22384:on 2013-03-13
22383:
22379:
22378:
22372:
22369:
22365:
22361:
22357:
22356:
22351:
22347:
22343:
22339:
22335:
22331:
22327:
22324:
22318:
22314:
22310:
22306:
22302:
22298:
22294:
22290:
22286:
22279:
22278:
22272:
22269:
22265:
22261:
22257:
22253:
22249:
22245:
22242:
22238:
22234:
22230:
22226:
22222:
22219:
22215:
22211:
22207:
22203:
22202:
22194:
22190:
22189:Massey, J. L.
22186:
22183:
22177:
22173:
22168:
22165:
22161:
22157:
22153:
22146:
22141:
22127:
22120:
22119:
22113:
22112:
22089:
22085:
22078:
22071:
22063:
22059:
22054:
22049:
22045:
22041:
22040:
22032:
22018:
22013:
22009:
22008:
22000:
21992:
21986:
21982:
21978:
21973:
21968:
21964:
21960:
21953:
21945:
21939:
21935:
21931:
21926:
21921:
21917:
21913:
21906:
21899:
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21890:
21885:
21881:
21877:
21876:
21868:
21859:
21852:
21847:
21838:
21831:
21816:
21812:
21808:
21801:
21794:
21779:
21775:
21769:
21762:
21756:
21748:
21744:
21740:
21736:
21732:
21728:
21721:
21714:
21712:
21703:
21699:
21695:
21693:9780470546345
21689:
21685:
21678:
21671:
21665:
21661:
21657:
21653:
21647:
21636:
21635:
21627:
21625:
21615:
21610:
21606:
21602:
21598:
21591:
21583:
21579:
21575:
21569:
21565:
21564:
21556:
21548:
21544:
21541:. MIT Press.
21540:
21533:
21525:
21521:
21517:
21513:
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21498:
21492:
21487:
21483:
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21256:
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21129:
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21109:
21104:
21100:
21092:
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21070:
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21063:
21055:
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21021:
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21006:
21002:
20978:
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20918:
20915:
20912:
20908:
20904:
20899:
20896:
20892:
20884:
20883:
20882:
20874:
20863:
20858:
20855:
20838:
20834:
20830:
20826:
20822:
20818:
20814:
20810:
20802:
20798:
20794:
20790:
20782:
20778:
20774:
20770:
20766:
20762:
20757:
20742:
20736:
20729:
20722:
20715:
20708:
20701:
20694:
20688:
20683:
20678:
20670:
20666:
20656:
20652:
20642:
20638:
20628:
20624:
20614:
20610:
20600:
20596:
20586:
20582:
20575:
20568:
20562:
20557:
20552:
20547:
20542:
20537:
20532:
20525:
20520:
20515:
20510:
20505:
20500:
20495:
20488:
20483:
20478:
20473:
20468:
20463:
20458:
20451:
20446:
20441:
20436:
20431:
20426:
20421:
20414:
20409:
20404:
20399:
20394:
20389:
20384:
20377:
20372:
20367:
20362:
20357:
20352:
20347:
20340:
20335:
20330:
20325:
20320:
20315:
20310:
20304:
20294:
20292:
20287:
20272:
20266:
20259:
20252:
20245:
20238:
20231:
20224:
20218:
20213:
20208:
20200:
20196:
20186:
20182:
20172:
20168:
20158:
20154:
20144:
20140:
20130:
20126:
20116:
20112:
20105:
20098:
20092:
20087:
20082:
20077:
20072:
20067:
20062:
20055:
20050:
20045:
20040:
20035:
20030:
20025:
20018:
20013:
20008:
20003:
19998:
19993:
19988:
19981:
19976:
19971:
19966:
19961:
19956:
19951:
19944:
19939:
19934:
19929:
19924:
19919:
19914:
19907:
19902:
19897:
19892:
19887:
19882:
19877:
19870:
19865:
19860:
19855:
19850:
19845:
19840:
19834:
19824:
19809:
19804:
19800:
19794:
19790:
19786:
19783:
19775:
19770:
19766:
19760:
19756:
19752:
19747:
19742:
19738:
19732:
19728:
19724:
19719:
19714:
19710:
19704:
19700:
19696:
19691:
19687:
19681:
19677:
19673:
19668:
19664:
19657:
19649:
19645:
19639:
19635:
19631:
19626:
19622:
19613:
19609:
19600:
19598:
19593:
19580:
19577:
19569:
19565:
19558:
19555:
19547:
19543:
19536:
19531:
19527:
19518:
19512:
19508:
19504:
19499:
19493:
19488:
19482:
19478:
19474:
19470:
19465:
19459:
19454:
19452:
19447:
19443:
19433:
19419:
19394:
19390:
19383:
19374:
19364:
19347:
19344:
19341:
19315:
19312:
19309:
19303:
19300:
19297:
19289:
19286:
19280:
19269:
19266:
19250:
19231:
19227:
19204:
19200:
19191:
19182:
18668:Syndrome_Vals
18635:Syndrome_Vals
17646:
17345:
17343:
17328:
17325:
17321:
17317:
17316:soft-decision
17313:
17309:
17305:
17297:Soft-decoding
17294:
17291:
17287:
17282:
17281:list decoding
17277:
17258:
17254:
17247:
17244:
17236:
17235:list-decoding
17232:
17227:
17223:
17218:
17214:
17210:
17202:
17196:
17190:
17186:
17181:
17177:
17173:
17169:
17159:
17157:
17153:
17150:) to produce
17149:
17145:
17141:
17137:
17133:
17129:
17125:
17121:
17116:
17099:
17096:
17093:
17090:
17087:
17072:
17069:
17066:
17062:
17056:
17048:
17045:
17042:
17037:
17034:
17031:
17027:
17021:
17013:
17008:
17005:
17002:
16998:
16992:
16981:
16978:
16976:
16969:
16965:
16952:
16948:
16942:
16939:
16936:
16928:
16925:
16922:
16917:
16914:
16911:
16907:
16901:
16893:
16888:
16884:
16873:
16865:
16860:
16857:
16855:
16848:
16844:
16817:
16813:
16790:
16786:
16777:
16773:
16769:
16764:
16748:
16744:
16721:
16717:
16707:
16694:
16691:
16688:
16685:
16682:
16668:
16664:
16657:
16654:
16649:
16645:
16641:
16636:
16632:
16628:
16623:
16619:
16610:
16606:
16602:
16598:
16594:
16590:
16586:
16582:
16578:
16573:
16571:
16567:
16563:
16559:
16555:
16551:
16547:
16543:
16539:
16535:
16531:
16527:
16523:
16519:
16515:
16511:
16507:
16503:
16499:
16495:
16491:
16487:
16483:
16479:
16475:
16471:
16467:
16463:
16459:
16455:
16451:
16447:
16435:
16428:
16424:
16417:
16413:
16406:
16402:
16392:
16388:
16384:
16381:
16377:
16374:
16373:
16369:
16365:
16362:
16358:
16354:
16351:
16350:
16346:
16343:
16339:
16335:
16331:
16328:
16327:
16323:
16320:
16316:
16312:
16308:
16304:
16301:
16300:
16296:
16291:
16288:
16283:
16281:
16278:
16277:
16274:
16266:
16259:
16255:
16248:
16244:
16239:
16235:
16231:
16223:
16219:
16214:
16210:
16206:
16201:
16198:
16194:
16190:
16186:
16182:
16174:
16170:
16167:
16161:
16157:
16152:
16148:
16143:
16139:
16136:
16130:
16126:
16121:
16117:
16112:
16108:
16102:
16098:
16094:
16090:
16086:
16082:
16077:
16073:
16069:
16065:
16058:
16051:
16047:
16043:
16036:
16033:
16026:
16022:
16020:
16016:
16012:
16008:
16002:
15998:
15993:
15989:
15983:
15979:
15974:
15970:
15964:
15960:
15955:
15951:
15947:
15945:
15941:
15936:
15932:
15928:
15925:decreases as
15924:
15917:
15912:
15908:
15904:
15900:
15895:
15891:
15887:
15883:
15879:
15874:
15870:
15866:
15863:
15860:
15845:
15837:
15825:
15820:
15806:
15802:
15796:
15784:
15777:
15770:
15761:
15757:
15751:
15747:
15737:
15733:
15727:
15723:
15713:
15709:
15703:
15699:
15692:
15687:
15682:
15674:
15670:
15662:
15655:
15651:
15645:
15637:
15632:
15628:
15618:
15614:
15606:
15602:
15596:
15588:
15583:
15579:
15573:
15565:
15560:
15556:
15546:
15542:
15534:
15530:
15524:
15516:
15511:
15507:
15501:
15493:
15488:
15484:
15478:
15470:
15465:
15461:
15451:
15447:
15439:
15435:
15429:
15421:
15416:
15412:
15406:
15398:
15393:
15389:
15383:
15375:
15370:
15366:
15356:
15352:
15344:
15340:
15334:
15326:
15321:
15317:
15311:
15303:
15298:
15294:
15288:
15280:
15275:
15271:
15261:
15257:
15249:
15245:
15239:
15231:
15226:
15222:
15216:
15208:
15203:
15199:
15193:
15179:
15175:
15167:
15163:
15157:
15149:
15144:
15140:
15134:
15120:
15116:
15108:
15104:
15098:
15087:
15077:
15073:
15068:
15052:
15043:
15038:
15034:
15027:
15021:
15018:
15012:
15006:
15000:
14985:
14966:
14958:
14955:
14950:
14942:
14939:
14936:
14931:
14928:
14925:
14921:
14915:
14912:
14909:
14901:
14896:
14892:
14886:
14878:
14876:
14868:
14855:
14852:
14849:
14844:
14836:
14833:
14830:
14825:
14822:
14819:
14815:
14809:
14806:
14803:
14795:
14790:
14786:
14780:
14772:
14770:
14762:
14747:
14743:
14739:
14736:
14731:
14727:
14723:
14720:
14717:
14712:
14709:
14706:
14702:
14696:
14693:
14690:
14686:
14682:
14677:
14674:
14671:
14667:
14661:
14657:
14653:
14651:
14643:
14637:
14625:
14621:
14617:
14613:
14609:
14605:
14600:
14598:
14588:
14586:
14582:
14572:
14569:
14566:
14562:
14559:
14555:
14551:
14548:
14545:
14542:
14541:
14537:
14534:
14531:
14527:
14524:
14520:
14516:
14513:
14510:
14507:
14506:
14502:
14499:
14496:
14493:
14489:
14486:
14483:
14480:
14479:
14475:
14472:
14469:
14466:
14462:
14459:
14456:
14453:
14452:
14449:
14446:
14444:
14441:
14439:
14436:
14434:
14431:
14429:
14426:
14422:
14418:
14415:
14413:
14410:
14409:
14406:
14398:
14396:
14392:
14388:
14384:
14380:
14376:
14358:
14355:
14352:
14348:
14339:
14331:
14328:
14325:
14320:
14317:
14314:
14310:
14301:
14293:
14288:
14284:
14280:
14269:
14265:
14261:
14251:
14249:
14245:
14241:
14223:
14219:
14215:
14212:
14207:
14203:
14199:
14196:
14191:
14187:
14181:
14177:
14173:
14168:
14164:
14158:
14154:
14108:
14106:
14101:
14095:
14087:
14080:
14075:
14060:
14054:
14047:
14041:
14036:
14031:
14023:
14009:
13998:
13991:
13985:
13980:
13973:
13968:
13962:
13952:
13947:
13932:
13926:
13919:
13913:
13908:
13903:
13897:
13894:
13887:
13884:
13878:
13873:
13868:
13860:
13846:
13835:
13828:
13822:
13817:
13810:
13805:
13799:
13778:
13775:
13767:
13763:
13756:
13753:
13748:
13744:
13739:
13736:
13733:
13725:
13721:
13714:
13711:
13706:
13702:
13697:
13694:
13691:
13683:
13679:
13672:
13669:
13664:
13660:
13640:
13637:
13634:
13631:
13628:
13625:
13622:
13619:
13614:
13610:
13606:
13603:
13600:
13595:
13591:
13587:
13584:
13581:
13576:
13572:
13568:
13565:
13562:
13557:
13553:
13549:
13546:
13543:
13538:
13534:
13530:
13527:
13524:
13516:
13512:
13505:
13502:
13497:
13493:
13484:
13481:at powers of
13480:
13464:
13461:
13458:
13455:
13452:
13447:
13443:
13439:
13436:
13431:
13427:
13423:
13420:
13415:
13411:
13407:
13404:
13399:
13395:
13391:
13388:
13383:
13379:
13375:
13372:
13366:
13360:
13357:
13351:
13345:
13342:
13336:
13330:
13310:
13307:
13304:
13301:
13298:
13293:
13289:
13285:
13282:
13277:
13273:
13269:
13266:
13261:
13257:
13253:
13250:
13245:
13241:
13237:
13234:
13229:
13225:
13221:
13218:
13212:
13204:
13200:
13196:
13191:
13187:
13179:
13173:
13170:
13164:
13158:
13139:
13136:
13133:
13130:
13127:
13122:
13118:
13114:
13111:
13106:
13102:
13098:
13095:
13089:
13081:
13071:
13067:
13059:
13053:
13050:
13044:
13036:
13032:
13021:
13017:
13013:
13009:
12992:
12989:
12986:
12983:
12980:
12975:
12971:
12967:
12964:
12959:
12955:
12951:
12948:
12943:
12939:
12935:
12927:
12923:
12919:
12916:
12905:
12901:
12897:
12894:
12883:
12879:
12875:
12872:
12863:
12860:
12857:
12851:
12845:
12839:
12831:
12825:
12818:
12811:
12800:
12798:
12794:
12790:
12786:
12782:
12772:
12765:
12761:
12751:
12749:
12730:
12722:
12709:
12707:
12703:
12699:
12692:
12679:
12677:
12659:
12656:
12651:
12647:
12624:
12620:
12611:
12603:
12592:
12590:
12586:
12581:
12577:
12573:
12555:
12547:
12544:
12541:
12537:
12533:
12526:
12517:
12514:
12511:
12507:
12503:
12494:
12491:
12488:
12484:
12480:
12474:
12469:
12464:
12456:
12444:
12435:
12432:
12429:
12415:
12404:
12397:
12389:
12386:
12383:
12380:
12376:
12370:
12363:
12360:
12357:
12353:
12345:
12341:
12333:
12328:
12323:
12318:
12309:
12306:
12303:
12299:
12293:
12286:
12282:
12274:
12270:
12260:
12256:
12250:
12243:
12239:
12231:
12227:
12220:
12209:
12204:
12200:
12196:
12191:
12176:
12173:
12170:
12166:
12162:
12159:
12154:
12144:
12141:
12138:
12135:
12132:
12128:
12124:
12121:
12118:
12113:
12110:
12107:
12097:
12094:
12091:
12087:
12083:
12078:
12068:
12064:
12041:
12038:
12035:
12031:
12021:
12008:
12005:
12000:
11996:
11990:
11982:
11977:
11974:
11971:
11967:
11961:
11958:
11955:
11947:
11944:
11941:
11936:
11933:
11930:
11927:
11924:
11920:
11914:
11906:
11901:
11898:
11895:
11891:
11881:
11868:
11865:
11861:
11855:
11850:
11846:
11840:
11836:
11830:
11825:
11822:
11819:
11815:
11810:
11804:
11796:
11793:
11790:
11786:
11780:
11777:
11774:
11771:
11768:
11763:
11759:
11753:
11749:
11743:
11738:
11735:
11732:
11728:
11723:
11717:
11709:
11705:
11699:
11696:
11693:
11690:
11687:
11682:
11678:
11672:
11668:
11662:
11657:
11654:
11651:
11647:
11642:
11636:
11628:
11624:
11618:
11615:
11612:
11607:
11603:
11597:
11593:
11587:
11582:
11579:
11576:
11572:
11567:
11537:
11524:
11521:
11517:
11511:
11506:
11502:
11496:
11492:
11486:
11476:
11471:
11468:
11465:
11461:
11456:
11452:
11449:
11446:
11442:
11436:
11433:
11430:
11427:
11424:
11419:
11415:
11409:
11405:
11399:
11389:
11384:
11381:
11378:
11374:
11369:
11365:
11361:
11355:
11352:
11349:
11346:
11343:
11338:
11334:
11328:
11324:
11318:
11308:
11303:
11300:
11297:
11293:
11288:
11284:
11280:
11274:
11271:
11268:
11263:
11259:
11253:
11249:
11243:
11238:
11235:
11232:
11228:
11223:
11213:
11200:
11197:
11193:
11187:
11182:
11178:
11172:
11168:
11162:
11154:
11151:
11148:
11143:
11140:
11137:
11134:
11131:
11126:
11122:
11116:
11112:
11106:
11098:
11093:
11090:
11087:
11084:
11081:
11076:
11072:
11066:
11062:
11056:
11048:
11043:
11040:
11037:
11032:
11028:
11022:
11018:
11013:
11007:
11002:
10999:
10996:
10992:
10983:
10979:
10974:
10957:
10954:
10949:
10944:
10940:
10934:
10930:
10924:
10916:
10913:
10910:
10905:
10902:
10899:
10896:
10893:
10888:
10884:
10878:
10874:
10868:
10860:
10855:
10852:
10849:
10846:
10843:
10838:
10834:
10828:
10824:
10818:
10810:
10805:
10802:
10799:
10794:
10790:
10784:
10780:
10771:
10768:
10763:
10760:
10755:
10751:
10745:
10742:
10739:
10734:
10730:
10724:
10720:
10714:
10706:
10703:
10700:
10695:
10692:
10687:
10683:
10677:
10674:
10671:
10666:
10662:
10656:
10652:
10646:
10638:
10633:
10630:
10625:
10621:
10615:
10612:
10609:
10604:
10600:
10594:
10590:
10584:
10576:
10571:
10568:
10565:
10560:
10556:
10550:
10546:
10537:
10534:
10530:
10524:
10521:
10516:
10512:
10506:
10498:
10495:
10492:
10487:
10484:
10479:
10475:
10469:
10461:
10456:
10453:
10448:
10444:
10438:
10430:
10427:
10423:
10417:
10414:
10411:
10406:
10402:
10396:
10392:
10383:
10380:
10372:
10369:
10364:
10360:
10348:
10345:
10342:
10337:
10333:
10327:
10323:
10295:
10292:
10289:
10284:
10280:
10274:
10270:
10249:
10246:
10243:
10240:
10237:
10217:
10208:
10195:
10192:
10184:
10181:
10176:
10172:
10145:
10142:
10139:
10136:
10133:
10130:
10122:
10118:
10114:
10109:
10106:
10101:
10097:
10093:
10090:
10065:
10062:
10057:
10053:
10049:
10046:
10024:
10021:
10016:
10012:
10001:
9996:The zeros of
9994:
9979:
9975:
9969:
9961:
9958:
9955:
9950:
9946:
9940:
9932:
9927:
9923:
9917:
9909:
9906:
9903:
9895:
9891:
9887:
9884:
9881:
9873:
9868:
9865:
9862:
9858:
9854:
9848:
9832:
9827:
9822:
9820:
9807:
9793:
9773:
9770:
9761:
9747:
9744:
9741:
9733:
9726:
9721:
9720:
9714:
9706:
9691:
9683:
9680:
9677:
9673:
9665:
9656:
9652:
9642:
9638:
9631:
9626:
9621:
9613:
9609:
9601:
9592:
9588:
9578:
9574:
9567:
9560:
9552:
9549:
9546:
9541:
9537:
9531:
9524:
9521:
9518:
9513:
9509:
9501:
9498:
9495:
9490:
9486:
9478:
9473:
9468:
9463:
9454:
9449:
9445:
9439:
9432:
9427:
9423:
9415:
9410:
9406:
9396:
9391:
9387:
9381:
9374:
9369:
9365:
9357:
9352:
9348:
9341:
9331:
9324:
9317:
9310:
9305:
9301:
9297:
9291:
9275:
9270:
9266:
9262:
9257:
9245:
9241:
9236:
9229:
9222:
9218:
9214:
9211:
9207:
9203:
9196:
9192:
9181:
9177:
9164:
9149:
9144:
9140:
9134:
9130:
9124:
9119:
9116:
9113:
9109:
9105:
9100:
9096:
9086:
9069:
9065:
9060:
9056:
9051:
9047:
9040:
9033:
9029:
9024:
9020:
9015:
9011:
9002:
8996:
8992:
8986:
8976:
8972:
8956:
8952:
8944:has roots at
8928:
8922:
8902:
8899:
8891:
8887:
8880:
8856:
8853:
8850:
8847:
8844:
8841:
8838:
8835:
8832:
8829:
8826:
8822:
8815:
8811:
8805:
8800:
8796:
8792:
8783:
8779:
8774:
8768:
8763:
8760:
8757:
8753:
8749:
8747:
8734:
8730:
8723:
8720:
8718:
8705:
8701:
8694:
8691:
8688:
8685:
8677:
8673:
8666:
8663:
8655:
8651:
8644:
8641:
8633:
8629:
8622:
8619:
8617:
8610:
8606:
8592:
8588:
8570:
8567:
8564:
8560:
8556:
8551:
8547:
8532:
8530:
8526:
8522:
8518:
8514:
8510:
8506:
8496:
8489:
8484:
8467:
8463:
8458:
8450:
8446:
8441:
8435:
8430:
8427:
8424:
8420:
8416:
8410:
8404:
8396:
8392:
8385:
8381:
8377:
8369:
8354:
8350:
8344:
8340:
8334:
8331:
8328:
8323:
8320:
8317:
8313:
8309:
8303:
8297:
8275:
8269:
8266:
8260:
8254:
8251:
8245:
8239:
8231:
8227:
8223:
8219:
8214:
8201:
8198:
8195:
8192:
8189:
8186:
8183:
8180:
8177:
8174:
8171:
8165:
8162:
8159:
8151:
8147:
8140:
8120:
8117:
8109:
8105:
8101:
8098:
8082:
8076:
8073:
8065:
8061:
8057:
8054:
8038:
8032:
8024:
8020:
8016:
8012:
8007:
8005:
7989:
7981:
7977:
7973:
7970:
7962:
7959:
7956:
7951:
7948:
7945:
7941:
7937:
7931:
7925:
7917:
7913:
7909:
7905:
7900:
7885:
7881:
7875:
7871:
7865:
7862:
7859:
7854:
7851:
7848:
7844:
7840:
7834:
7828:
7820:
7816:
7795:
7792:
7789:
7785:
7781:
7778:
7775:
7770:
7766:
7762:
7759:
7756:
7751:
7747:
7729:
7727:
7723:
7716:
7706:
7698:
7696:
7691:
7687:
7678:
7676:
7672:
7656:
7648:
7630:
7626:
7622:
7617:
7613:
7584:
7580:
7573:
7570:
7567:
7564:
7556:
7552:
7545:
7519:
7516:
7510:
7504:
7476:
7473:
7470:
7466:
7459:
7456:
7453:
7450:
7442:
7438:
7431:
7408:
7388:
7363:
7355:
7352:
7329:
7326:
7323:
7320:
7317:
7314:
7311:
7305:
7302:
7280:
7276:
7255:
7235:
7228:of the field
7227:
7211:
7202:
7197:
7195:
7191:
7187:
7182:
7180:
7175:
7158:
7155:
7152:
7146:
7140:
7137:
7134:
7111:
7108:
7105:
7085:
7082:
7079:
7071:
7055:
7052:
7049:
7029:
7021:
7020:byte-oriented
7005:
7002:
6999:
6996:
6991:
6987:
6983:
6980:
6960:
6957:
6952:
6948:
6944:
6941:
6921:
6899:
6895:
6874:
6865:
6851:
6831:
6808:
6805:
6800:
6796:
6791:
6786:
6783:
6761:
6753:
6750:
6747:
6741:
6738:
6735:
6730:
6726:
6702:
6699:
6690:
6681:
6678:
6673:
6670:
6648:
6645:
6642:
6632:
6628:
6624:
6621:
6613:
6608:
6604:
6592:
6589:
6578:
6573:
6568:
6565:
6562:
6559:
6556:
6552:
6546:
6543:
6536:
6533:
6528:
6523:
6519:
6496:
6493:
6490:
6480:
6476:
6472:
6469:
6461:
6456:
6452:
6440:
6437:
6426:
6421:
6416:
6413:
6410:
6407:
6404:
6400:
6394:
6391:
6383:
6380:
6375:
6371:
6364:
6361:
6358:
6354:
6348:
6343:
6339:
6330:
6326:
6317:
6313:
6299:
6279:
6270:
6266:
6262:
6258:
6252:
6248:
6244:
6241:
6225:
6221:
6214:
6211:
6208:
6185:
6182:
6179:
6170:
6168:
6164:
6160:
6156:
6140:
6137:
6134:
6131:
6128:
6125:
6116:
6108:
6104:
6101:
6097:
6093:
6089:
6079:
6066:
6059:
6053:
6048:
6044:
6041:
6035:
6027:
6023:
6019:
6013:
6005:
6001:
5997:
5991:
5983:
5979:
5975:
5970:
5966:
5962:
5956:
5950:
5947:
5941:
5935:
5912:
5906:
5861:
5855:
5846:
5833:
5826:
5818:
5814:
5810:
5805:
5801:
5797:
5791:
5785:
5782:
5776:
5770:
5747:
5741:
5721:
5698:
5692:
5670:
5666:
5662:
5656:
5650:
5628:
5624:
5620:
5615:
5611:
5607:
5604:
5601:
5596:
5593:
5590:
5586:
5582:
5577:
5574:
5571:
5567:
5546:
5543:
5540:
5531:
5518:
5512:
5506:
5488:
5484:
5480:
5474:
5468:
5465:
5459:
5451:
5447:
5423:
5415:
5411:
5390:
5367:
5361:
5341:
5338:
5335:
5332:
5329:
5307:
5303:
5279:
5273:
5264:
5247:
5241:
5218:
5212:
5189:
5183:
5160:
5154:
5131:
5125:
5102:
5096:
5073:
5067:
5047:
5024:
5018:
4995:
4989:
4983:
4977:
4974:
4968:
4962:
4954:
4944:
4931:
4927:
4921:
4918:
4915:
4911:
4907:
4904:
4901:
4896:
4892:
4888:
4883:
4879:
4868:
4864:
4858:
4854:
4848:
4843:
4840:
4837:
4833:
4829:
4823:
4817:
4804:
4798:
4794:
4790:
4787:
4784:
4779:
4775:
4771:
4766:
4762:
4757:
4752:
4748:
4735:
4721:
4718:
4715:
4706:
4691:
4688:
4685:
4681:
4677:
4672:
4669:
4666:
4663:
4660:
4656:
4650:
4647:
4644:
4641:
4638:
4634:
4630:
4627:
4624:
4621:
4616:
4612:
4608:
4603:
4599:
4595:
4591:
4585:
4582:
4579:
4576:
4573:
4570:
4567:
4563:
4559:
4556:
4552:
4548:
4544:
4538:
4535:
4532:
4528:
4524:
4521:
4517:
4512:
4506:
4502:
4498:
4495:
4491:
4487:
4481:
4475:
4456:
4433:
4427:
4407:
4404:
4401:
4378:
4372:
4365:
4349:
4346:
4343:
4320:
4314:
4291:
4285:
4262:
4256:
4236:
4233:
4230:
4227:
4224:
4204:
4201:
4198:
4175:
4169:
4146:
4140:
4126:
4124:
4118:
4100:
4097:
4094:
4090:
4086:
4083:
4080:
4075:
4071:
4067:
4064:
4061:
4058:
4052:
4047:
4044:
4041:
4037:
4033:
4030:
4027:
4022:
4018:
3996:
3992:
3988:
3983:
3979:
3970:
3966:
3962:
3958:
3953:
3938:
3927:
3924:
3921:
3917:
3908:
3904:
3896:
3884:
3880:
3871:
3867:
3854:
3850:
3841:
3837:
3830:
3825:
3819:
3813:
3791:
3787:
3778:
3768:
3752:
3748:
3725:
3722:
3719:
3715:
3711:
3708:
3705:
3700:
3696:
3670:
3667:
3664:
3660:
3651:
3647:
3643:
3640:
3637:
3629:
3625:
3616:
3612:
3591:
3583:
3578:
3563:
3552:
3549:
3546:
3542:
3533:
3529:
3521:
3509:
3505:
3496:
3492:
3479:
3475:
3466:
3462:
3455:
3450:
3444:
3438:
3430:
3414:
3411:
3408:
3404:
3400:
3397:
3394:
3389:
3385:
3376:
3360:
3338:
3334:
3313:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3283:
3280:
3270:
3266:
3262:
3254:
3250:
3241:
3237:
3216:
3194:
3190:
3167:
3163:
3154:
3144:
3130:
3122:
3118:
3102:
3094:
3089:
3074:
3066:
3063:
3060:
3056:
3048:
3039:
3035:
3025:
3021:
3014:
3007:
2999:
2996:
2993:
2988:
2985:
2982:
2978:
2972:
2965:
2960:
2957:
2954:
2950:
2942:
2939:
2936:
2932:
2926:
2919:
2914:
2909:
2904:
2899:
2890:
2887:
2884:
2879:
2875:
2869:
2862:
2857:
2853:
2845:
2841:
2835:
2826:
2823:
2820:
2815:
2811:
2805:
2798:
2793:
2789:
2781:
2777:
2771:
2765:
2760:
2757:
2754:
2751:
2745:
2739:
2719:
2699:
2679:
2676:
2673:
2653:
2650:
2647:
2641:
2635:
2627:
2611:
2589:
2578:
2575:
2572:
2568:
2559:
2555:
2547:
2535:
2531:
2522:
2518:
2505:
2501:
2492:
2488:
2481:
2476:
2470:
2464:
2442:
2438:
2429:
2425:
2421:
2418:
2398:
2391:of the field
2376:
2373:
2370:
2366:
2362:
2359:
2356:
2351:
2347:
2326:
2304:
2300:
2279:
2259:
2253:
2249:
2243:
2239:
2233:
2230:
2227:
2222:
2219:
2216:
2212:
2208:
2202:
2194:
2190:
2167:
2163:
2140:
2136:
2132:
2124:
2121:
2118:
2114:
2110:
2107:
2104:
2099:
2095:
2088:
2085:
2077:
2067:
2065:
2061:
2057:
2039:
2035:
2031:
2028:
2025:
2020:
2016:
2007:
2003:
1999:
1995:
1991:
1987:
1969:
1965:
1956:
1951:
1949:
1933:
1929:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1899:
1895:
1879:
1875:
1871:
1868:
1865:
1845:
1842:
1839:
1836:
1833:
1829:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1803:
1799:
1796:
1793:
1789:
1785:
1782:
1779:
1776:
1773:
1769:
1765:
1762:
1759:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1713:
1697:
1694:
1691:
1671:
1668:
1665:
1662:
1659:
1656:
1650:
1647:
1644:
1638:
1635:
1615:
1612:
1609:
1589:
1581:
1565:
1553:
1550:
1542:
1534:
1510:
1506:
1499:
1496:
1493:
1490:
1482:
1478:
1471:
1468:
1460:
1456:
1449:
1431:
1395:
1393:
1389:
1385:
1381:
1377:
1373:
1369:
1365:
1361:
1357:
1353:
1350:− 1}, {0, 1,
1349:
1345:
1342:of the field
1327:
1323:
1319:
1316:
1313:
1308:
1304:
1295:
1291:
1287:
1271:
1251:
1243:
1227:
1207:
1199:
1189:
1175:
1172:
1169:
1166:
1163:
1143:
1140:
1137:
1115:
1112:
1107:
1104:
1097:
1093:
1077:
1057:
1037:
1030:
1014:
1011:
1008:
1005:
1002:
994:
979:
972:
956:
949:
933:
925:
910:
908:
905:
902:
901:
897:
895:
892:
889:
888:
884:
881:
878:
877:
873:
870:
867:
866:
862:
859:
856:
855:
851:
848:
845:
844:
840:
838:
835:
832:
831:
827:
824:
821:
818:
817:
813:
810:
807:
804:
803:
799:
796:
793:
792:
788:
785:
782:
781:
775:
773:
768:
766:
762:
758:
754:
750:
746:
741:
739:
735:
733:
729:
724:
722:
714:
709:
700:
698:
695:as a form of
694:
690:
686:
681:
679:
675:
671:
667:
662:
660:
656:
652:
648:
644:
640:
636:
632:
622:
620:
615:
613:
609:
605:
601:
597:
587:
585:
581:
577:
572:
569:
565:
563:
559:
556:
555:convolutional
552:
548:
544:
539:
526:
524:
519:
517:
516:
510:
508:
503:
501:
497:
493:
490:
489:convolutional
486:
482:
478:
474:
470:
466:
462:
458:
454:
446:
442:
440:
435:
433:
429:
425:
420:
418:
414:
409:
406:developed by
405:
400:
397:
393:
389:
385:
381:
377:
373:
363:
361:
356:
344:
339:
331:
325:
319:
308: −
299:
294:
292:
288:
284:
280:
276:
272:
268:
264:
260:
256:
252:
248:
244:
240:
236:
232:
228:
217:
212:
210:
205:
203:
198:
197:
194:
191:
187:
182:
179:
175:
171:
167:
162:
157:
153:
151:
147:
143:
139:
133:
129:
125:
121:
118:
116:
115:Alphabet size
112:
108:
104:
101:
99:
95:
92:
89:
87:
83:
80:
77:
75:
71:
66:
62:
59:
55:
50:
47:
43:
40:
36:
31:
19:
22501:
22476:
22422:, Springer,
22418:
22395:
22382:the original
22376:
22359:
22353:
22341:
22337:
22312:
22292:
22276:
22259:
22255:
22232:
22228:
22199:
22171:
22155:
22151:
22133:, retrieved
22126:the original
22117:
22095:. Retrieved
22083:
22070:
22043:
22037:
22031:
22021:, retrieved
22006:
21999:
21962:
21952:
21915:
21905:
21879:
21873:
21867:
21858:
21850:
21846:
21837:
21830:compact disc
21822:, retrieved
21810:
21806:
21793:
21782:. Retrieved
21768:
21755:
21730:
21726:
21683:
21677:
21655:
21646:
21633:
21607:(1): 87–99.
21604:
21600:
21590:
21562:
21555:
21538:
21532:
21507:
21503:
21497:
21486:
21462:
21433:Chien search
21404:
21396:
21389:
21385:
21378:
21374:
21372:Recalculate
21371:
21365:
21361:
21357:
21353:
21349:
21345:
21341:
21337:
21329:
21325:
21321:
21317:
21309:
21305:
21301:
21297:
21293:
21289:
21282:
21278:
21274:
21270:
21266:
21262:
21260:
21254:
21250:
21243:
21239:
21235:
21227:
21223:
21219:
21215:
21208:
21204:
21200:
21182:152 x + 237
21146:
21138:
21131:
21051:
21047:
20880:
20872:
20861:
20853:
20843:Recalculate
20842:
20836:
20832:
20828:
20824:
20820:
20816:
20812:
20808:
20800:
20796:
20792:
20788:
20780:
20776:
20772:
20768:
20764:
20760:
20295:
20288:
19825:
19601:
19596:
19594:
19519:
19516:
19510:
19506:
19502:
19497:
19491:
19486:
19480:
19476:
19472:
19468:
19463:
19457:
19450:
19445:
19441:
19439:
19370:
19188:
19179:
18614:% magnitudes
17666:% Example:
17644:
17339:
17300:
17289:
17285:
17278:
17219:
17212:
17208:
17200:
17194:
17188:
17184:
17179:
17175:
17171:
17165:
17155:
17151:
17147:
17143:
17139:
17135:
17131:
17127:
17123:
17119:
17117:
16775:
16771:
16767:
16765:
16708:
16608:
16604:
16600:
16596:
16592:
16588:
16584:
16580:
16576:
16574:
16569:
16565:
16561:
16557:
16553:
16549:
16545:
16541:
16537:
16533:
16529:
16525:
16521:
16517:
16513:
16509:
16505:
16501:
16497:
16493:
16489:
16485:
16481:
16477:
16473:
16469:
16465:
16461:
16457:
16453:
16449:
16445:
16443:
16433:
16432:/ 544 = 546
16426:
16422:
16415:
16411:
16410:/ 544 = 329
16404:
16400:
16390:
16386:
16379:
16367:
16360:
16356:
16341:
16337:
16333:
16318:
16314:
16310:
16306:
16294:
16286:
16279:
16272:
16257:
16253:
16252:
16246:
16242:
16237:
16233:
16229:
16221:
16217:
16212:
16208:
16204:
16196:
16192:
16188:
16184:
16180:
16178:
16172:
16168:
16165:
16159:
16155:
16150:
16146:
16141:
16137:
16134:
16128:
16124:
16119:
16115:
16110:
16106:
16100:
16096:
16092:
16088:
16084:
16080:
16075:
16071:
16067:
16063:
16056:
16049:
16045:
16041:
16034:
16031:
16024:
16018:
16014:
16010:
16006:
16005:
16000:
15996:
15991:
15987:
15981:
15977:
15972:
15968:
15962:
15958:
15953:
15949:
15943:
15939:
15934:
15930:
15926:
15922:
15920:
15915:
15910:
15906:
15902:
15898:
15893:
15889:
15885:
15881:
15877:
15872:
15868:
15864:
15861:
15075:
15071:
15069:
14986:
14623:
14619:
14615:
14611:
14607:
14603:
14601:
14594:
14584:
14580:
14578:
14564:
14557:
14553:
14529:
14522:
14518:
14491:
14464:
14447:
14442:
14437:
14432:
14427:
14420:
14416:
14411:
14404:
14394:
14390:
14386:
14378:
14374:
14267:
14263:
14257:
14247:
14243:
14239:
14146:Subtracting
14145:
14102:
14099:
14093:
14085:
14078:
13948:
13482:
13478:
13019:
13015:
13011:
13007:
12823:
12816:
12809:
12806:
12796:
12792:
12788:
12784:
12774:
12767:
12763:
12759:
12757:
12744:
12717:
12715:
12694:
12687:
12685:
12676:Chien search
12606:
12601:
12598:
12584:
12579:
12575:
12571:
12207:
12202:
12198:
12194:
12193:Recall that
12192:
12023:Subtracting
12022:
11882:
11538:
11214:
10981:
10977:
10975:
10209:
9999:
9995:
9830:
9825:
9823:
9815:
9813:
9762:
9728:
9725:erasure code
9722:
9716:
9709:
9707:
9326:
9319:
9312:
9308:
9303:
9299:
9295:
9292:
9165:
9087:
8997:
8995:error values
8994:
8987:
8984:
8982:
8973:
8590:
8586:
8538:
8528:
8524:
8520:
8516:
8512:
8508:
8498:
8491:
8487:
8485:
8394:
8387:
8383:
8379:
8372:
8371:Coefficient
8370:
8229:
8225:
8221:
8217:
8215:
8022:
8018:
8014:
8010:
8008:
8003:
7915:
7911:
7907:
7903:
7901:
7818:
7814:
7735:
7721:
7718:
7704:
7692:
7688:
7684:
7198:
7193:
7183:
7176:
7069:
6866:
6327:channel for
6322:
6268:
6264:
6260:
6256:
6171:
6158:
6154:
6105:and minimum
6102:
6095:
6091:
6085:
5847:
5532:
5265:
4950:
4736:
4707:
4132:
4119:
3968:
3964:
3960:
3956:
3954:
3774:
3579:
3431:
3150:
3090:
2073:
2059:
2055:
2005:
2001:
1997:
1994:coefficients
1993:
1989:
1954:
1952:
1897:
1579:
1396:
1391:
1383:
1379:
1375:
1371:
1367:
1363:
1359:
1355:
1351:
1347:
1343:
1293:
1289:
1285:
1241:
1195:
1070:, and thus,
1029:finite field
992:
948:block length
921:
890:2004–present
879:1996–present
846:1977–present
833:1977–present
794:1958–present
769:
742:
736:
728:concatenated
725:
718:
687:systems and
682:
677:
676:is usually 2
673:
669:
665:
663:
658:
654:
650:
646:
642:
628:
616:
593:
573:
570:
566:
543:compact disc
540:
537:
534:Data storage
529:Applications
520:
513:
511:
504:
471:devices and
463:, where two
461:compact disc
450:
436:
428:James Massey
421:
417:cyclic codes
401:
391:
387:
369:
357:
342:
330:erasure code
323:
317:
298:finite-field
295:
226:
225:
177:
155:
141:
137:
131:
127:
123:
119:
106:
102:
90:
78:
74:Block length
22493:Dave Forney
22484:, TM-102162
22303:, AD0669824
21438:Cyclic code
21394:to correct
20869:Gao decoder
20851:to correct
17653:= rsDecoder
17552:k, n, alpha
17312:turbo codes
17304:demodulator
17222:Madhu Sudan
16066: := 0
16062: := 1
16055: := 0
9824:Define the
8133:Therefore,
7732:Formulation
6240:demodulator
4123:Gao decoder
3117:linear code
1092:prime power
894:Turbo codes
789:Mission(s)
772:turbo codes
635:Vandermonde
621:symbology.
558:interleaver
465:interleaved
263:Data Matrix
38:Named after
22558:Categories
22097:2017-06-07
22023:2024-02-08
22017:2304.09445
21972:2304.01403
21925:2206.05256
21824:2009-09-28
21784:2019-02-01
21660:IEEE Press
21478:References
18983:decoded_gf
18932:decoded_gf
18701:size_error
18680:size_error
18617:size_error
18344:max_errors
17672:max_errors
17519:msg_padded
17498:msg_padded
17459:msg_padded
16575:Transform
16070:degree of
12716:Calculate
8873:Note that
7695:puncturing
6090:of length
6082:Properties
4394:of degree
4191:of degree
3582:systematic
3229:such that
3153:systematic
1362:}, or for
1090:must be a
907:LDPC codes
612:Aztec Code
604:Datamatrix
492:inner code
184:Properties
164:Algorithms
22338:SIAM News
22311:(1984) ,
22135:April 21,
22048:CiteSeerX
21884:CiteSeerX
21702:557445046
21640:, Clemson
21547:859669631
21068:−
20935:−
20909:∏
20897:−
19787:−
19658:−
19556:−
19301:−
19099:prim_poly
18959:errors_gf
18923:prim_poly
18899:errors_gf
18884:error_mag
18869:error_pos
18848:prim_poly
18821:prim_poly
18794:error_mag
18734:error_pos
18641:Syndromes
18629:error_pos
18593:error_mag
18575:orig_vals
18563:error_pos
18512:error_pos
18401:remainder
18326:remainder
18119:prim_poly
18077:prim_poly
18053:Syndromes
18005:prim_poly
17927:error_mag
17918:error_pos
17900:orig_vals
17882:Syndromes
17852:Syndromes
17795:prim_poly
17705:orig_vals
17525:remainder
17477:prim_poly
17414:prim_poly
17356:rsEncoder
17220:In 1999,
17084:for
17070:−
17053:Λ
17046:⋯
17035:−
17018:Λ
17006:−
16989:Λ
16982:−
16940:−
16933:Λ
16926:⋯
16915:−
16898:Λ
16870:Λ
16861:−
16692:≤
16686:≤
16679:for
16665:α
16524:). Since
16154: :=
16123: :=
16095: :=
16087: :=
16040: :=
16030: :=
15946:/2, then
15834:Ω
15817:Ω
15793:Ω
15642:Λ
15593:Λ
15570:Λ
15521:Λ
15498:Λ
15475:Λ
15426:Λ
15403:Λ
15380:Λ
15331:Λ
15308:Λ
15285:Λ
15236:Λ
15213:Λ
15190:Λ
15154:Λ
15131:Λ
15095:Λ
15047:Ω
14995:Λ
14963:Ω
14947:Ω
14940:⋯
14929:−
14913:−
14906:Ω
14883:Ω
14863:Ω
14841:Λ
14834:⋯
14823:−
14807:−
14800:Λ
14777:Λ
14757:Λ
14721:⋯
14710:−
14694:−
14675:−
14614:), and Ω(
14356:−
14336:Λ
14329:⋯
14318:−
14298:Λ
14278:Δ
14091:= 562 = 3
14020:Λ
14006:Λ
13895:−
13885:−
13857:Λ
13843:Λ
13626:⋅
13607:⋅
13588:⋅
13569:⋅
13550:⋅
13531:⋅
13197:−
12920:−
12898:−
12876:−
12861:−
12758:Finally,
12731:α
12657:−
12589:Gill n.d.
12548:ν
12542:ν
12534:−
12527:⋮
12512:ν
12504:−
12489:ν
12481:−
12453:Λ
12445:⋮
12433:−
12430:ν
12426:Λ
12416:ν
12412:Λ
12387:−
12384:ν
12371:⋯
12358:ν
12346:ν
12334:⋮
12329:⋱
12324:⋮
12319:⋮
12304:ν
12294:⋯
12261:ν
12251:⋯
12177:ν
12163:−
12151:Λ
12142:−
12139:ν
12122:⋯
12111:−
12108:ν
12104:Λ
12079:ν
12075:Λ
12042:ν
11991:ν
11987:Λ
11959:−
11956:ν
11952:Λ
11945:⋯
11934:−
11931:ν
11911:Λ
11902:ν
11831:ν
11816:∑
11805:ν
11801:Λ
11794:⋯
11778:−
11775:ν
11744:ν
11729:∑
11714:Λ
11697:−
11694:ν
11663:ν
11648:∑
11633:Λ
11619:ν
11588:ν
11573:∑
11547:Λ
11487:ν
11483:Λ
11477:ν
11462:∑
11450:⋯
11434:−
11431:ν
11396:Λ
11390:ν
11375:∑
11353:−
11350:ν
11315:Λ
11309:ν
11294:∑
11275:ν
11244:ν
11229:∑
11163:ν
11159:Λ
11152:⋯
11141:−
11138:ν
11103:Λ
11091:−
11088:ν
11053:Λ
11044:ν
11008:ν
10993:∑
10925:ν
10921:Λ
10914:⋯
10903:−
10900:ν
10865:Λ
10853:−
10850:ν
10815:Λ
10806:ν
10764:ν
10761:−
10746:ν
10715:ν
10711:Λ
10704:⋯
10693:−
10678:ν
10643:Λ
10631:−
10616:ν
10581:Λ
10572:ν
10525:ν
10522:−
10507:ν
10503:Λ
10496:⋯
10485:−
10466:Λ
10454:−
10435:Λ
10418:ν
10370:−
10354:Λ
10349:ν
10296:ν
10250:ν
10247:≤
10241:≤
10182:−
10166:Λ
10137:−
10115:⋅
10107:−
10094:−
10063:−
10022:−
9980:ν
9970:ν
9966:Λ
9959:⋯
9937:Λ
9914:Λ
9885:−
9874:ν
9859:∏
9843:Λ
9745:−
9681:−
9666:⋮
9614:ν
9602:⋮
9550:−
9542:ν
9532:⋯
9522:−
9499:−
9479:⋮
9474:⋱
9469:⋮
9464:⋮
9450:ν
9440:⋯
9392:ν
9382:⋯
9237:α
9215:∗
9208:α
9178:α
9125:ν
9110:∑
9025:α
8953:α
8888:α
8854:−
8845:…
8797:α
8769:ν
8754:∑
8731:α
8702:α
8674:α
8652:α
8630:α
8568:−
8561:α
8557:…
8548:α
8436:ν
8421:∑
8332:−
8314:∑
8199:−
8190:…
8148:α
8106:α
8102:−
8062:α
8058:−
7978:α
7974:−
7960:−
7942:∏
7863:−
7845:∑
7793:−
7779:…
7760:…
7627:α
7614:α
7581:α
7568:…
7553:α
7517:α
7508:↦
7474:−
7467:α
7454:…
7439:α
7327:−
7318:…
7306:∈
7277:α
7212:α
6997:−
6958:−
6806:
6751:−
6742:−
6700:−
6649:ℓ
6646:−
6625:−
6614:ℓ
6593:ℓ
6579:ℓ
6557:ℓ
6553:∑
6529:≈
6497:ℓ
6494:−
6473:−
6462:ℓ
6441:ℓ
6427:ℓ
6405:ℓ
6401:∑
6381:−
6362:−
6349:≈
6212:−
6183:−
6132:−
6100:dimension
6042:≡
6020:−
5998:≡
5976:−
5963:⋅
5948:≡
5811:−
5798:⋅
5663:⋅
5605:…
5594:−
5575:−
5544:−
5481:⋅
5339:−
4919:−
4912:α
4905:…
4893:α
4880:α
4834:∑
4788:…
4689:−
4670:−
4664:−
4648:−
4642:−
4628:⋯
4583:−
4577:−
4564:α
4560:−
4549:⋯
4529:α
4525:−
4503:α
4499:−
4457:α
4405:−
4362:, with a
4347:−
4234:−
4228:≤
4202:−
4098:−
4091:α
4084:…
4072:α
4065:α
4045:−
4031:…
3993:α
3925:−
3897:⋯
3723:−
3709:…
3668:−
3641:…
3604:entries,
3550:−
3522:⋯
3412:−
3398:…
3305:−
3296:…
3284:∈
3064:−
3049:⋮
2997:−
2986:−
2973:…
2958:−
2940:−
2920:⋮
2915:⋱
2910:⋮
2905:⋮
2900:⋮
2888:−
2870:…
2824:−
2806:…
2677:×
2576:−
2548:⋯
2435:→
2374:−
2360:…
2231:−
2213:∑
2133:∈
2122:−
2108:…
2029:…
1917:≤
1908:δ
1843:−
1837:∼
1817:−
1783:−
1760:δ
1731:−
1695:−
1663:−
1648:−
1639:−
1613:−
1494:…
1317:…
1173:−
1050:of order
1012:≤
903:est. 2009
868:1989–2003
857:1989–2003
819:1969–1975
805:1968–1978
404:BCH codes
279:broadcast
243:MiniDiscs
174:Euclidean
57:Hierarchy
22393:(1977),
22332:(1993),
22301:45195002
22291:(1967),
22191:(1969),
22088:Archived
21815:archived
21778:Archived
21582:45727875
21418:BCH code
21412:See also
21360:) = 003
21324:) = 001
21296:) = 003
20831:) = 003
20795:) = 001
20767:) = 003
19475:) = 003
19412:, where
19111:function
19001:function
18530:evalPoly
18458:evalPoly
18233:quotient
18200:quotient
18188:quotient
17650:function
17540:function
17349:function
17187:−
16736:through
16187:) and Ω(
15074:= 6 and
14626:errors:
10976:Sum for
8915:because
7728:(1961).
7605:; since
7345:, where
7190:inverted
6267:−
6251:MDS code
6247:erasures
2692:-matrix
1858:, where
1712:distance
1580:distinct
969:, and a
924:alphabet
874:Galileo
863:Voyager
841:Voyager
633:-RS and
600:MaxiCode
590:Bar code
576:parchive
396:BCH code
259:QR codes
150:Notation
130: (
98:Distance
22491:by Dr.
22218:9003708
21747:9289140
21524:2098821
21504:J. SIAM
21261:divide
21050:= 1 to
20877:Example
19436:Example
18977:decoded
18938:encoded
18569:decoded
18557:isempty
18464:polyval
18107:size_r0
18029:size_r0
17894:decoded
17876:isempty
17819:encoded
17813:polyval
17711:encoded
17641:Decoder
17547:genpoly
17513:encoded
17429:genpoly
17351:encoded
17336:Encoder
17288:−
17211:−
16516:), and
16492:), and
16269:Example
15980:) = −Q(
15942:) <
14602:Define
14401:Example
14143:) = 122
14128:) = 074
12803:Example
12700:in the
10980:= 1 to
8397:, then
7681:Remarks
7042:, with
6157:,
6098:) with
2008:points
1992:as the
1378:, ...,
1358:, ...,
825:(32, 6)
797:Uncoded
757:Cassini
749:Galileo
645:,
619:PostBar
608:QR Code
596:PDF-417
366:History
257:discs,
255:Blu-ray
22426:
22403:
22319:
22299:
22216:
22178:
22050:
21987:
21940:
21886:
21745:
21700:
21690:
21666:
21580:
21570:
21545:
21522:
21383:where
21364:+ 002
21328:+ 924
21308:+ 007
21304:+ 009
21300:+ 916
21269:) and
21253:+ 176
21249:= 708
21226:+ 311
21222:+ 798
21218:+ 086
21214:= 266
20847:where
20835:+ 002
20799:+ 924
20779:+ 007
20775:+ 009
20771:+ 916
20289:Using
19479:+ 002
19134:length
18911:errors
18863:errors
18623:length
18602:return
18281:length
18206:length
18167:deconv
18035:length
17957:return
17726:errors
17492:deconv
17342:MATLAB
17215:) / 2⌋
16603:) and
16468:) and
16414:+ 821
16393:+ 544
16389:+ 704
16382:+ 596
16370:+ 396
16363:+ 024
16359:+ 676
16344:+ 732
16340:+ 637
16336:+ 762
16321:+ 000
16317:+ 000
16313:+ 000
16309:+ 000
16091:+ 1
16013:) and
15999:) = Ω(
15961:) = Λ(
14618:) for
14556:+ 821
14521:+ 173
14377:) and
14345:
14307:
14139:)/Λ'(x
14135:= −Ω(x
14124:)/Λ'(x
14120:= −Ω(x
13949:Using
13014:) = 3
12830:PDF417
9044:
8169:
8009:Since
8002:where
7671:cyclic
7201:cyclic
7179:bursts
6663:where
6094:(over
5502:
5494:
4217:where
2002:values
1370:, {1,
631:Cauchy
610:, and
580:USENET
500:DVB-S2
291:RAID 6
178:et al.
136:Often
134:prime)
22506:(PDF)
22450:(CMU)
22281:(PDF)
22214:S2CID
22196:(PDF)
22148:(PDF)
22129:(PDF)
22122:(PDF)
22091:(PDF)
22080:(PDF)
22012:arXiv
21967:arXiv
21920:arXiv
21818:(PDF)
21803:(PDF)
21743:S2CID
21723:(PDF)
21638:(PDF)
21520:JSTOR
21454:Notes
21368:+ 001
21332:+ 006
21312:+ 006
21257:+ 532
21230:+ 532
20845:P(x)
20839:+ 001
20803:+ 006
20783:+ 006
19483:+ 001
18854:'
18710:alpha
18476:alpha
18359:break
18155:while
18095:zeros
17825:alpha
17771:alpha
17732:zeros
17678:floor
17447:alpha
17390:alpha
16583:) to
16436:+ 732
16418:+ 001
16200:(0):
16191:) by
16083:/2
16068:while
15078:= 3:
14610:), Λ(
12814:with
12812:(929)
7401:over
7224:is a
7068:, of
6887:with
3959:<
3353:from
3095:over
2624:is a
2182:with
1898:every
1386:is a
1366:<
1264:with
995:with
926:size
783:Years
730:with
689:CCSDS
584:Wuala
485:DVB-S
338:burst
275:WiMAX
159:-code
22482:NASA
22424:ISBN
22401:ISBN
22317:ISBN
22297:OCLC
22233:IT-6
22176:ISBN
22137:2010
21985:ISBN
21938:ISBN
21759:See
21698:OCLC
21688:ISBN
21664:ISBN
21578:OCLC
21568:ISBN
21543:OCLC
21352:) =
21344:) /
21242:) =
21207:) =
21171:001
21160:000
21046:for
20823:) =
20815:) /
19453:− 1
19158:xpad
19152:<
19123:x, k
19113:xpad
19054:find
19008:trim
18524:find
18443:trim
18395:trim
18377:trim
18251:trim
18227:conv
18158:true
18017:trim
17951:Synd
17864:Synd
17858:trim
17807:Synd
17617:conv
17310:and
17308:LDPC
17224:and
17166:The
17134:) −
17126:) =
17097:<
17091:<
16766:Let
16709:Use
16564:) +
16556:) =
16540:) +
16532:) =
16452:) =
16425:) =
16403:) =
16385:608
16378:673
16366:697
16355:683
16347:001
16332:925
16324:000
16305:001
16232:) =
16207:) =
15888:) +
15070:For
14563:173
14552:329
14528:173
14517:634
14490:173
14463:197
14258:The
13018:+ 2
12821:and
12773:and
10210:Let
9003:as:
8993:and
7295:for
7070:data
7053:<
6325:AWGN
1551:<
1096:rate
1006:<
946:, a
786:Code
755:and
685:xDSL
562:CIRC
549:and
496:LDPC
426:and
374:and
287:ATSC
285:and
273:and
251:DVDs
237:and
144:− 1.
44:and
22364:doi
22344:(1)
22264:doi
22237:doi
22206:doi
22160:doi
22084:QEX
22058:doi
21977:doi
21930:doi
21894:doi
21735:doi
21609:doi
21512:doi
21154:−1
20737:003
20730:916
20723:009
20716:007
20709:006
20702:924
20695:006
20563:001
20558:000
20553:000
20548:000
20543:000
20538:000
20533:000
20526:000
20521:001
20516:000
20511:000
20506:000
20501:000
20496:000
20489:000
20484:000
20479:001
20474:000
20469:000
20464:000
20459:000
20452:000
20447:000
20442:000
20437:001
20432:000
20427:000
20422:000
20415:000
20410:000
20405:000
20400:000
20395:001
20390:000
20385:000
20378:000
20373:000
20368:000
20363:000
20358:000
20353:001
20348:000
20341:000
20336:000
20331:000
20326:000
20321:000
20316:000
20311:001
20267:289
20260:637
20253:017
20246:541
20239:437
20232:923
20225:000
20093:562
20088:713
20083:893
20078:923
20073:928
20068:726
20063:121
20056:304
20051:804
20046:904
20041:924
20036:928
20031:430
20026:086
20019:673
20014:865
20009:913
20004:925
19999:928
19994:228
19989:057
19982:848
19977:902
19972:920
19967:926
19962:928
19957:439
19952:456
19945:913
19940:921
19935:925
19930:927
19925:928
19920:246
19915:123
19908:928
19903:928
19898:928
19893:928
19888:928
19883:006
19878:006
19871:000
19866:000
19861:000
19856:000
19851:928
19846:000
19841:001
19348:249
19342:255
19219:to
19170:end
19167:end
19149:len
19128:len
19118:pad
19105:end
19075:end
18995:end
18788:end
18782:err
18770:idx
18746:err
18743:));
18719:idx
18689:idx
18686:for
18656:(:,
18608:end
18509:));
18431:end
18365:end
18338:end
18320:all
18290:));
18269:pad
18215:));
18194:pad
17963:end
17849:));
17635:end
17632:end
17581:mod
17572:for
17531:end
17504:g_x
17423:g_x
17191:+ 1
17158:).
16805:to
16572:).
16508:),
16484:),
16302:−1
16250:(0)
16225:(0)
14587:).
14570:412
14567:+ 1
14560:+ 1
14549:576
14546:925
14535:412
14532:+ 1
14525:+ 1
14514:412
14511:762
14500:732
14494:+ 1
14487:846
14484:637
14473:732
14467:+ 1
14460:732
14457:732
14397:).
14216:122
14055:821
14048:329
13986:001
13981:000
13974:000
13969:001
13927:004
13920:167
13898:925
13888:762
13823:762
13818:637
13811:637
13806:732
13779:925
13737:762
13695:637
13641:732
13635:474
13623:487
13604:191
13585:456
13566:123
13465:474
13456:487
13440:191
13424:456
13408:123
13311:474
13302:487
13286:191
13270:382
13140:455
13131:442
13115:738
13099:547
13078:mod
13022:+ 1
12993:522
12984:568
12968:723
12952:809
12826:= 4
12819:= 3
12743:of
12674:).
9835:as
9302:≥ 2
8531:).
8393:of
8232:).
8090:mod
8046:mod
7918:):
7821:):
7724:by
7159:223
7153:255
7112:255
7086:223
7006:255
6797:log
6695:min
6329:FSK
6121:min
6049:mod
5498:mod
5295:by
3123:is
2319:at
1390:of
1156:or
691:'s
551:DVD
547:DAT
360:BCH
326:/2⌋
283:DVB
271:DSL
247:CDs
109:+ 1
22560::
22360:11
22358:,
22342:26
22340:,
22336:,
22258:,
22250:;
22231:,
22212:,
22198:,
22156:43
22154:,
22150:,
22082:.
22056:.
22044:49
22042:.
22010:,
21983:.
21975:.
21961:.
21936:.
21928:.
21914:.
21892:,
21880:45
21878:,
21811:51
21809:,
21805:,
21776:.
21741:.
21731:95
21729:.
21725:.
21710:^
21696:.
21662:,
21658:,
21623:^
21605:27
21603:.
21599:.
21576:.
21518:.
21506:.
21470:).
21381:)
21187:2
21176:1
21165:0
20293::
19509:+
19505:=
19449:=
19251:,
19146:if
19143:);
19102:);
19078:),
19060:gx
19048:gx
19042:gf
19036:gt
19018:gx
19003:gt
18974:);
18926:);
18905:gf
18851:))
18830:gf
18809:er
18803:gf
18776::)
18764:er
18713:.^
18683:);
18632:);
18590:);
18554:if
18545:);
18539:),
18533:==
18518:gf
18479:.^
18452:);
18419:g1
18413:g1
18407:g0
18404:);
18389:r1
18386:);
18383:r1
18371:r0
18350:==
18317:if
18302:g0
18287:g0
18260:);
18242:);
18239:g1
18212:g1
18182:);
18179:r1
18173:r0
18143:f0
18137:g1
18131:f1
18125:g0
18122:);
18110:),
18089:gf
18083:f1
18080:);
18068:(,
18065:gf
18059:f0
18047:r1
18044:);
18041:r0
18026:);
18023:r0
18011:r0
18008:);
17993:r0
17987:gf
17981:r0
17972:r0
17915:);
17873:if
17867:);
17828:.^
17798:);
17777:gf
17747:);
17702:);
17681:((
17629:);
17507:);
17480:);
17468:(,
17465:gf
17450:);
17417:);
17396:gf
17207:⌊(
17199:⌊(
16421:Ω(
16399:Λ(
16375:2
16352:1
16329:0
16265:.
16241:/
16228:Ω(
16216:/
16203:Λ(
16175:-1
16164:-
16162:-2
16144:-1
16133:-
16131:-2
16113:-1
16105:/
16103:-2
16079:≥
16053:−1
16048:)
16028:−1
16003:).
15905:=
15901:)
15880:)
14599:.
14573:2
14538:1
14503:2
14476:1
14423:+1
14270::
14250:.
14200:74
14107:.
13953::
13485:.
12810:GF
12708:.
10958:0.
10772:0.
10538:0.
10384:0.
9998:Λ(
9829:Λ(
9821:.
9298:−
9290:.
7677:.
6824:,
6776:,
6718:,
6331::
6263:≤
6259:+
6169:.
6141:1.
5928::
5763::
5439::
4734:.
4125:.
3971::
3775:A
3767:.
3429:.
3143:.
2732::
1950:.
1896:,
1394:.
1374:,
1354:,
1292:≤
1188:.
774::
767:.
761:dB
751:,
747:,
723:.
713:CC
699:.
606:,
602:,
598:,
525:.
518:.
502:.
441:.
434:.
419:.
318:or
304:=
293:.
277:,
265:,
261:,
253:,
249:,
245:,
140:=
126:≥
122:=
105:−
22366::
22266::
22260:8
22239::
22208::
22162::
22100:.
22064:.
22060::
22014::
21993:.
21979::
21969::
21946:.
21932::
21922::
21896::
21832:.
21787:.
21749:.
21737::
21704:.
21617:.
21611::
21584:.
21549:.
21526:.
21514::
21508:9
21405:c
21397:b
21390:x
21388:(
21386:E
21379:x
21377:(
21375:P
21366:x
21362:x
21358:x
21356:(
21354:P
21350:x
21348:(
21346:E
21342:x
21340:(
21338:Q
21330:x
21326:x
21322:x
21320:(
21318:E
21310:x
21306:x
21302:x
21298:x
21294:x
21292:(
21290:Q
21283:x
21281:(
21279:E
21275:x
21273:(
21271:E
21267:x
21265:(
21263:Q
21255:x
21251:x
21247:2
21244:A
21240:x
21238:(
21236:E
21228:x
21224:x
21220:x
21216:x
21212:2
21209:R
21205:x
21203:(
21201:Q
21147:i
21144:A
21139:i
21136:R
21132:i
21113:1
21110:=
21105:0
21101:A
21079:0
21076:=
21071:1
21064:A
21052:n
21048:i
21034:}
21031:)
21026:i
21022:a
21018:(
21015:b
21012:,
21007:i
21003:a
20999:{
20979:=
20974:0
20970:R
20948:)
20943:i
20939:a
20932:x
20929:(
20924:n
20919:1
20916:=
20913:i
20905:=
20900:1
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20862:c
20854:b
20837:x
20833:x
20829:x
20827:(
20825:P
20821:x
20819:(
20817:E
20813:x
20811:(
20809:Q
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20793:x
20791:(
20789:E
20781:x
20777:x
20773:x
20769:x
20765:x
20763:(
20761:Q
20743:]
20689:[
20684:=
20679:]
20671:4
20667:q
20657:3
20653:q
20643:2
20639:q
20629:1
20625:q
20615:0
20611:q
20601:1
20597:e
20587:0
20583:e
20576:[
20569:]
20305:[
20273:]
20219:[
20214:=
20209:]
20201:4
20197:q
20187:3
20183:q
20173:2
20169:q
20159:1
20155:q
20145:0
20141:q
20131:1
20127:e
20117:0
20113:e
20106:[
20099:]
19835:[
19810:2
19805:i
19801:a
19795:i
19791:b
19784:=
19781:)
19776:4
19771:i
19767:a
19761:4
19757:q
19753:+
19748:3
19743:i
19739:a
19733:3
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19725:+
19720:2
19715:i
19711:a
19705:2
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19661:(
19655:)
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19640:1
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19632:+
19627:0
19623:e
19619:(
19614:i
19610:b
19597:e
19581:0
19578:=
19575:)
19570:i
19566:a
19562:(
19559:Q
19553:)
19548:i
19544:a
19540:(
19537:E
19532:i
19528:b
19511:e
19507:c
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19477:x
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19384:O
19351:)
19345:,
19339:(
19316:!
19313:k
19310:!
19307:)
19304:k
19298:n
19295:(
19290:!
19287:n
19281:=
19275:)
19270:k
19267:n
19262:(
19232:n
19228:a
19205:1
19201:a
19164:;
19161:=
19155:k
19140:x
19137:(
19131:=
19125:)
19121:(
19115:=
19096:.
19093:g
19090:,
19087:m
19084:.
19081:g
19072::
19069:)
19066:1
19063:,
19057:(
19051:(
19045:(
19039:=
19033:;
19030:x
19027:.
19024:g
19021:=
19015:)
19013:g
19011:(
19005:=
18992:;
18989:x
18986:.
18980:=
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18968::
18965:1
18962:(
18956:+
18953:)
18950:k
18947::
18944:1
18941:(
18935:=
18920:,
18917:m
18914:,
18908:(
18902:=
18893:;
18890:x
18887:.
18881:=
18878:)
18875:x
18872:.
18866:(
18857:;
18845:,
18842:m
18839:,
18836:b
18833:(
18827:\
18824:)
18818:,
18815:m
18812:,
18806:(
18800:(
18797:=
18785:;
18779:=
18773:,
18767:(
18761:;
18758:x
18755:.
18752:e
18749:=
18740:x
18737:.
18731:-
18728:n
18725:(
18722:*
18716:(
18707:=
18704:e
18698::
18695:1
18692:=
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18674:1
18671:(
18665:=
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18659:1
18653:b
18650:;
18647:x
18644:.
18638:=
18626:(
18620:=
18605:;
18599:;
18596:=
18587:k
18584::
18581:1
18578:(
18572:=
18566:)
18560:(
18542:m
18536:0
18527:(
18521:(
18515:=
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18503::
18500:1
18497:-
18494::
18491:1
18488:-
18485:n
18482:(
18473:,
18470:g
18467:(
18461:=
18449:g
18446:(
18440:=
18437:g
18428:;
18425:g
18422:=
18416:;
18410:=
18398:(
18392:=
18380:(
18374:=
18362:;
18356:)
18353:0
18347:)
18341:-
18335::
18332:1
18329:(
18323:(
18311:;
18308:c
18305:-
18299:=
18296:g
18284:(
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18275:c
18272:(
18266:=
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18257:c
18254:(
18248:=
18245:c
18236:,
18230:(
18224:=
18221:c
18209:(
18203:,
18197:(
18191:=
18176:,
18170:(
18164:=
18146:;
18140:=
18134:;
18128:=
18116:,
18113:m
18104:,
18101:1
18098:(
18092:(
18086:=
18074:,
18071:m
18062:=
18056:;
18050:=
18038:(
18032:=
18020:(
18014:=
18002:,
17999:m
17996:,
17990:(
17984:=
17978:;
17975:=
17960:;
17954:;
17948:=
17945:S
17942:;
17939:=
17936:g
17933:;
17930:=
17924:;
17921:=
17912:k
17909::
17906:1
17903:(
17897:=
17891:)
17888:x
17885:.
17879:(
17861:(
17855:=
17846:k
17843:-
17840:n
17837::
17834:1
17831:(
17822:,
17816:(
17810:=
17792:,
17789:m
17786:,
17783:2
17780:(
17774:=
17765:;
17762:=
17759:S
17756:;
17753:=
17750:g
17744:n
17741:,
17738:1
17735:(
17729:=
17720:;
17717:x
17714:.
17708:=
17699:2
17696:/
17693:)
17690:k
17687:-
17684:n
17675:=
17660:)
17656:(
17626:,
17623:g
17620:(
17614:=
17611:g
17608:)
17605:n
17602:,
17599:k
17596:-
17593:n
17590::
17587:1
17584:(
17578:=
17575:k
17566:;
17563:1
17560:=
17557:g
17554:)
17550:(
17544:=
17542:g
17528:;
17522:-
17516:=
17501:,
17495:(
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17474:,
17471:m
17462:=
17444:,
17441:n
17438:,
17435:k
17432:(
17426:=
17411:,
17408:m
17405:,
17402:2
17399:(
17393:=
17363:)
17359:(
17353:=
17290:k
17286:n
17264:)
17259:m
17255:2
17251:(
17248:F
17245:G
17213:k
17209:n
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17156:x
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17146:(
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17140:x
17138:(
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17132:x
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17100:n
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17088:t
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17073:v
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17038:2
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17028:E
17022:2
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17009:1
17003:j
16999:E
16993:1
16985:(
16979:=
16970:j
16966:E
16958:)
16953:1
16949:E
16943:1
16937:v
16929:+
16923:+
16918:1
16912:v
16908:E
16902:1
16894:+
16889:v
16885:E
16881:(
16874:v
16866:1
16858:=
16849:0
16845:E
16818:t
16814:E
16791:1
16787:E
16776:x
16774:(
16772:E
16768:v
16749:t
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16722:1
16718:R
16695:t
16689:j
16683:1
16674:)
16669:j
16661:(
16658:r
16655:=
16650:j
16646:S
16642:=
16637:j
16633:E
16629:=
16624:j
16620:R
16609:x
16607:(
16605:E
16601:x
16599:(
16597:R
16593:t
16589:x
16587:(
16585:R
16581:x
16579:(
16577:r
16570:x
16568:(
16566:E
16562:x
16560:(
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16546:x
16544:(
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16536:(
16534:c
16530:x
16528:(
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16522:x
16520:(
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16514:x
16512:(
16510:e
16506:x
16504:(
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16494:R
16490:x
16488:(
16486:E
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16480:(
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16466:x
16464:(
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16342:x
16338:x
16334:x
16319:x
16315:x
16311:x
16307:x
16295:i
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16234:R
16230:x
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16218:A
16213:i
16209:A
16205:x
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16189:x
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16173:i
16169:A
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16160:i
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16142:i
16138:R
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16129:i
16125:R
16120:i
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16101:i
16097:R
16093:Q
16089:i
16085:i
16081:t
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16064:i
16060:0
16057:A
16050:A
16046:x
16044:(
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16038:0
16035:R
16032:x
16025:R
16019:x
16017:(
16015:Q
16011:x
16009:(
16007:B
16001:x
15997:x
15995:(
15992:i
15988:R
15984:)
15982:x
15978:x
15976:(
15973:i
15969:B
15965:)
15963:x
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15957:(
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15938:(
15935:i
15931:R
15927:i
15923:R
15918:)
15916:x
15914:(
15911:i
15907:R
15903:x
15899:x
15897:(
15894:i
15890:B
15886:x
15884:(
15882:S
15878:x
15876:(
15873:i
15869:A
15846:]
15838:0
15826:x
15821:1
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15785:0
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15758:x
15752:0
15748:Q
15738:7
15734:x
15728:1
15724:Q
15714:8
15710:x
15704:2
15700:Q
15693:[
15688:=
15683:]
15675:1
15671:S
15663:x
15656:1
15652:S
15646:1
15638:+
15633:2
15629:S
15619:2
15615:x
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15597:2
15589:+
15584:2
15580:S
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15525:3
15517:+
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15479:1
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15466:4
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15327:+
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15312:2
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15299:5
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15232:+
15227:5
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15217:2
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15158:3
15150:+
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15141:S
15135:2
15121:8
15117:x
15109:6
15105:S
15099:3
15088:[
15076:e
15072:t
15056:)
15053:x
15050:(
15044:+
15039:t
15035:x
15031:)
15028:x
15025:(
15022:Q
15019:=
15016:)
15013:x
15010:(
15007:S
15004:)
15001:x
14998:(
14967:0
14959:+
14956:x
14951:1
14943:+
14937:+
14932:1
14926:e
14922:x
14916:1
14910:e
14902:+
14897:e
14893:x
14887:e
14879:=
14872:)
14869:x
14866:(
14856:1
14853:+
14850:x
14845:1
14837:+
14831:+
14826:1
14820:e
14816:x
14810:1
14804:e
14796:+
14791:e
14787:x
14781:e
14773:=
14766:)
14763:x
14760:(
14748:1
14744:S
14740:+
14737:x
14732:2
14728:S
14724:+
14718:+
14713:2
14707:t
14703:x
14697:1
14691:t
14687:S
14683:+
14678:1
14672:t
14668:x
14662:t
14658:S
14654:=
14647:)
14644:x
14641:(
14638:S
14624:e
14620:t
14616:x
14612:x
14608:x
14606:(
14604:S
14585:x
14581:C
14565:x
14558:x
14554:x
14543:3
14530:x
14523:x
14519:x
14508:2
14497:1
14492:x
14481:1
14470:1
14465:x
14454:0
14448:m
14443:b
14438:B
14433:C
14428:d
14421:n
14417:S
14412:n
14395:x
14391:x
14389:(
14387:C
14379:e
14375:x
14359:e
14353:i
14349:S
14340:e
14332:+
14326:+
14321:1
14315:i
14311:S
14302:1
14294:+
14289:i
14285:S
14281:=
14268:e
14264:x
14248:s
14244:x
14242:(
14240:r
14224:4
14220:x
14213:+
14208:3
14204:x
14197:=
14192:4
14188:x
14182:2
14178:e
14174:+
14169:3
14165:x
14159:1
14155:e
14141:2
14137:2
14133:2
14131:e
14126:1
14122:1
14118:1
14116:e
14089:2
14086:x
14082:1
14079:x
14061:]
14042:[
14037:=
14032:]
14024:1
14010:2
13999:[
13992:]
13963:[
13933:]
13914:[
13909:=
13904:]
13879:[
13874:=
13869:]
13861:1
13847:2
13836:[
13829:]
13800:[
13776:=
13773:)
13768:4
13764:3
13760:(
13757:r
13754:=
13749:4
13745:S
13740:,
13734:=
13731:)
13726:3
13722:3
13718:(
13715:r
13712:=
13707:3
13703:S
13698:,
13692:=
13689:)
13684:2
13680:3
13676:(
13673:r
13670:=
13665:2
13661:S
13638:=
13632:+
13629:3
13620:+
13615:2
13611:3
13601:+
13596:3
13592:3
13582:+
13577:4
13573:3
13563:+
13558:5
13554:3
13547:2
13544:+
13539:6
13535:3
13528:3
13525:=
13522:)
13517:1
13513:3
13509:(
13506:r
13503:=
13498:1
13494:S
13483:α
13479:r
13462:+
13459:x
13453:+
13448:2
13444:x
13437:+
13432:3
13428:x
13421:+
13416:4
13412:x
13405:+
13400:5
13396:x
13392:2
13389:+
13384:6
13380:x
13376:3
13373:=
13370:)
13367:x
13364:(
13361:e
13358:+
13355:)
13352:x
13349:(
13346:s
13343:=
13340:)
13337:x
13334:(
13331:r
13308:+
13305:x
13299:+
13294:2
13290:x
13283:+
13278:3
13274:x
13267:+
13262:4
13258:x
13254:1
13251:+
13246:5
13242:x
13238:2
13235:+
13230:6
13226:x
13222:3
13219:=
13216:)
13213:x
13210:(
13205:r
13201:s
13192:t
13188:x
13183:)
13180:x
13177:(
13174:p
13171:=
13168:)
13165:x
13162:(
13159:s
13137:+
13134:x
13128:+
13123:2
13119:x
13112:+
13107:3
13103:x
13096:=
13093:)
13090:x
13087:(
13082:g
13072:t
13068:x
13063:)
13060:x
13057:(
13054:p
13051:=
13048:)
13045:x
13042:(
13037:r
13033:s
13020:x
13016:x
13012:x
13010:(
13008:p
12990:+
12987:x
12981:+
12976:2
12972:x
12965:+
12960:3
12956:x
12949:+
12944:4
12940:x
12936:=
12933:)
12928:4
12924:3
12917:x
12914:(
12911:)
12906:3
12902:3
12895:x
12892:(
12889:)
12884:2
12880:3
12873:x
12870:(
12867:)
12864:3
12858:x
12855:(
12852:=
12849:)
12846:x
12843:(
12840:g
12824:t
12817:α
12797:x
12795:(
12793:s
12789:x
12787:(
12785:r
12779:k
12777:i
12775:e
12770:k
12768:i
12764:x
12762:(
12760:e
12747:k
12745:X
12720:k
12718:i
12697:k
12695:Y
12690:k
12688:X
12660:1
12652:k
12648:X
12625:k
12621:X
12609:k
12607:X
12602:i
12585:ν
12580:ν
12576:ν
12572:ν
12556:]
12545:+
12538:S
12518:2
12515:+
12508:S
12495:1
12492:+
12485:S
12475:[
12470:=
12465:]
12457:1
12436:1
12405:[
12398:]
12390:1
12381:2
12377:S
12364:1
12361:+
12354:S
12342:S
12310:1
12307:+
12300:S
12287:3
12283:S
12275:2
12271:S
12257:S
12244:2
12240:S
12232:1
12228:S
12221:[
12208:i
12203:v
12199:v
12195:j
12174:+
12171:j
12167:S
12160:=
12155:1
12145:1
12136:+
12133:j
12129:S
12125:+
12119:+
12114:1
12098:1
12095:+
12092:j
12088:S
12084:+
12069:j
12065:S
12039:+
12036:j
12032:S
12009:0
12006:=
12001:j
11997:S
11983:+
11978:1
11975:+
11972:j
11968:S
11962:1
11948:+
11942:+
11937:1
11928:+
11925:j
11921:S
11915:1
11907:+
11899:+
11896:j
11892:S
11869:0
11866:=
11862:)
11856:j
11851:k
11847:X
11841:k
11837:Y
11826:1
11823:=
11820:k
11811:(
11797:+
11791:+
11787:)
11781:2
11772:+
11769:j
11764:k
11760:X
11754:k
11750:Y
11739:1
11736:=
11733:k
11724:(
11718:2
11710:+
11706:)
11700:1
11691:+
11688:j
11683:k
11679:X
11673:k
11669:Y
11658:1
11655:=
11652:k
11643:(
11637:1
11629:+
11625:)
11616:+
11613:j
11608:k
11604:X
11598:k
11594:Y
11583:1
11580:=
11577:k
11568:(
11525:0
11522:=
11518:)
11512:j
11507:k
11503:X
11497:k
11493:Y
11472:1
11469:=
11466:k
11457:(
11453:+
11447:+
11443:)
11437:2
11428:+
11425:j
11420:k
11416:X
11410:k
11406:Y
11400:2
11385:1
11382:=
11379:k
11370:(
11366:+
11362:)
11356:1
11347:+
11344:j
11339:k
11335:X
11329:k
11325:Y
11319:1
11304:1
11301:=
11298:k
11289:(
11285:+
11281:)
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9632:[
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8301:(
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8264:)
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8252:=
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8228:(
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8220:(
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8175:=
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8160:=
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8118:=
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8094:(
8086:)
8083:x
8080:(
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8074:=
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8055:x
8050:(
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8039:x
8036:(
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8021:(
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8015:x
8013:(
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7990:,
7987:)
7982:j
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7968:(
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7949:=
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7938:=
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7929:(
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7914:(
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7908:x
7906:(
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7886:i
7882:x
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7841:=
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7835:x
7832:(
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7817:(
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7801:)
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7776:,
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7757:,
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7744:(
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7631:1
7623:=
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7593:)
7590:)
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7571:,
7565:,
7562:)
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7435:(
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7368:|
7364:F
7360:|
7356:=
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7333:}
7330:1
7324:q
7321:,
7315:,
7312:1
7309:{
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7162:)
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7083:=
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7050:k
7030:k
7003:=
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6988:2
6984:=
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6961:1
6953:m
6949:2
6945:=
6942:n
6922:m
6900:m
6896:2
6875:F
6852:M
6832:s
6809:M
6801:2
6792:m
6787:=
6784:h
6762:h
6758:)
6754:s
6748:1
6745:(
6739:1
6736:=
6731:s
6727:P
6706:)
6703:1
6691:d
6687:(
6682:2
6679:1
6674:=
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6639:)
6633:s
6629:P
6622:1
6619:(
6609:s
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6598:)
6590:n
6585:(
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6566:+
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6560:=
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6544:1
6537:m
6534:1
6524:b
6520:P
6491:n
6487:)
6481:s
6477:P
6470:1
6467:(
6457:s
6453:P
6446:)
6438:n
6433:(
6422:n
6417:1
6414:+
6411:t
6408:=
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6384:1
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6372:2
6365:1
6359:m
6355:2
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6340:P
6300:S
6280:E
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6255:2
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6222:/
6218:)
6215:k
6209:n
6206:(
6186:k
6180:n
6159:k
6155:n
6138:+
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6129:n
6126:=
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6103:k
6096:F
6092:n
6067:.
6063:)
6060:x
6057:(
6054:g
6045:0
6039:)
6036:x
6033:(
6028:r
6024:s
6017:)
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5995:)
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5989:(
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5971:t
5967:x
5960:)
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5954:(
5951:p
5945:)
5942:x
5939:(
5936:s
5916:)
5913:x
5910:(
5907:g
5886:C
5865:)
5862:x
5859:(
5856:s
5834:.
5830:)
5827:x
5824:(
5819:r
5815:s
5806:t
5802:x
5795:)
5792:x
5789:(
5786:p
5783:=
5780:)
5777:x
5774:(
5771:s
5751:)
5748:x
5745:(
5742:p
5722:k
5702:)
5699:x
5696:(
5693:s
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5667:x
5660:)
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5654:(
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5629:0
5625:x
5621:,
5616:1
5612:x
5608:,
5602:,
5597:2
5591:t
5587:x
5583:,
5578:1
5572:t
5568:x
5547:1
5541:t
5519:.
5516:)
5513:x
5510:(
5507:g
5489:t
5485:x
5478:)
5475:x
5472:(
5469:p
5466:=
5463:)
5460:x
5457:(
5452:r
5448:s
5427:)
5424:x
5421:(
5416:r
5412:s
5391:t
5371:)
5368:x
5365:(
5362:g
5342:k
5336:n
5333:=
5330:t
5308:t
5304:x
5283:)
5280:x
5277:(
5274:p
5251:)
5248:x
5245:(
5242:p
5222:)
5219:x
5216:(
5213:s
5193:)
5190:x
5187:(
5184:p
5164:)
5161:x
5158:(
5155:g
5135:)
5132:x
5129:(
5126:s
5106:)
5103:x
5100:(
5097:s
5077:)
5074:x
5071:(
5068:p
5048:k
5028:)
5025:x
5022:(
5019:s
4999:)
4996:x
4993:(
4990:g
4987:)
4984:x
4981:(
4978:p
4975:=
4972:)
4969:x
4966:(
4963:s
4932:.
4928:}
4922:k
4916:n
4908:,
4902:,
4897:2
4889:,
4884:1
4869:i
4865:a
4859:i
4855:s
4849:n
4844:1
4841:=
4838:i
4830:=
4827:)
4824:a
4821:(
4818:s
4812:|
4805:)
4799:n
4795:s
4791:,
4785:,
4780:2
4776:s
4772:,
4767:1
4763:s
4758:(
4753:{
4749:=
4745:C
4722:1
4719:=
4716:i
4692:k
4686:n
4682:x
4678:+
4673:1
4667:k
4661:n
4657:x
4651:1
4645:k
4639:n
4635:g
4631:+
4625:+
4622:x
4617:1
4613:g
4609:+
4604:0
4600:g
4596:=
4592:)
4586:1
4580:k
4574:n
4571:+
4568:i
4557:x
4553:(
4545:)
4539:1
4536:+
4533:i
4522:x
4518:(
4513:)
4507:i
4496:x
4492:(
4488:=
4485:)
4482:x
4479:(
4476:g
4437:)
4434:x
4431:(
4428:g
4408:k
4402:n
4382:)
4379:x
4376:(
4373:g
4350:1
4344:k
4324:)
4321:x
4318:(
4315:p
4295:)
4292:x
4289:(
4286:s
4266:)
4263:x
4260:(
4257:s
4237:1
4231:q
4225:n
4205:1
4199:n
4179:)
4176:x
4173:(
4170:s
4150:)
4147:x
4144:(
4141:p
4106:}
4101:1
4095:n
4087:,
4081:,
4076:2
4068:,
4062:,
4059:1
4056:{
4053:=
4048:1
4042:n
4038:a
4034:,
4028:,
4023:0
4019:a
3997:i
3989:=
3984:i
3980:a
3969:α
3965:k
3961:q
3957:n
3939:]
3933:)
3928:1
3922:n
3918:a
3914:(
3909:m
3905:p
3890:)
3885:1
3881:a
3877:(
3872:m
3868:p
3860:)
3855:0
3851:a
3847:(
3842:m
3838:p
3831:[
3826:=
3823:)
3820:m
3817:(
3814:C
3792:m
3788:p
3753:m
3749:p
3726:1
3720:k
3716:m
3712:,
3706:,
3701:0
3697:m
3676:)
3671:1
3665:k
3661:a
3657:(
3652:m
3648:p
3644:,
3638:,
3635:)
3630:0
3626:a
3622:(
3617:m
3613:p
3592:k
3564:]
3558:)
3553:1
3547:n
3543:a
3539:(
3534:m
3530:p
3515:)
3510:1
3506:a
3502:(
3497:m
3493:p
3485:)
3480:0
3476:a
3472:(
3467:m
3463:p
3456:[
3451:=
3448:)
3445:m
3442:(
3439:C
3415:1
3409:n
3405:a
3401:,
3395:,
3390:k
3386:a
3361:m
3339:m
3335:p
3314:.
3311:}
3308:1
3302:k
3299:,
3293:,
3290:0
3287:{
3281:i
3271:i
3267:m
3263:=
3260:)
3255:i
3251:a
3247:(
3242:m
3238:p
3217:k
3195:m
3191:p
3168:m
3164:p
3131:A
3103:F
3075:]
3067:1
3061:k
3057:m
3040:1
3036:m
3026:0
3022:m
3015:[
3008:]
3000:1
2994:k
2989:1
2983:n
2979:a
2966:2
2961:1
2955:n
2951:a
2943:1
2937:n
2933:a
2927:1
2891:1
2885:k
2880:1
2876:a
2863:2
2858:1
2854:a
2846:1
2842:a
2836:1
2827:1
2821:k
2816:0
2812:a
2799:2
2794:0
2790:a
2782:0
2778:a
2772:1
2766:[
2761:=
2758:m
2755:A
2752:=
2749:)
2746:m
2743:(
2740:C
2720:F
2700:A
2680:k
2674:n
2654:m
2651:A
2648:=
2645:)
2642:m
2639:(
2636:C
2612:C
2590:]
2584:)
2579:1
2573:n
2569:a
2565:(
2560:m
2556:p
2541:)
2536:1
2532:a
2528:(
2523:m
2519:p
2511:)
2506:0
2502:a
2498:(
2493:m
2489:p
2482:[
2477:=
2474:)
2471:m
2468:(
2465:C
2443:n
2439:F
2430:k
2426:F
2422::
2419:C
2399:F
2377:1
2371:n
2367:a
2363:,
2357:,
2352:0
2348:a
2327:n
2305:m
2301:p
2280:m
2260:.
2254:i
2250:a
2244:i
2240:m
2234:1
2228:k
2223:0
2220:=
2217:i
2209:=
2206:)
2203:a
2200:(
2195:m
2191:p
2168:m
2164:p
2141:k
2137:F
2130:)
2125:1
2119:k
2115:m
2111:,
2105:,
2100:0
2096:m
2092:(
2089:=
2086:m
2060:k
2056:p
2040:k
2036:a
2032:,
2026:,
2021:1
2017:a
2006:k
1998:p
1990:x
1970:k
1966:q
1955:k
1934:n
1930:/
1926:1
1923:+
1920:1
1914:R
1911:+
1880:n
1876:/
1872:k
1869:=
1866:R
1846:R
1840:1
1834:n
1830:/
1826:1
1823:+
1820:R
1814:1
1811:=
1808:n
1804:/
1800:1
1797:+
1794:n
1790:/
1786:k
1780:1
1777:=
1774:n
1770:/
1766:d
1763:=
1740:1
1737:+
1734:k
1728:n
1725:=
1722:d
1698:1
1692:k
1672:1
1669:+
1666:k
1660:n
1657:=
1654:)
1651:1
1645:k
1642:(
1636:n
1616:1
1610:k
1590:k
1566:.
1560:}
1554:k
1543:F
1535:p
1529:|
1521:)
1516:)
1511:n
1507:a
1503:(
1500:p
1497:,
1491:,
1488:)
1483:2
1479:a
1475:(
1472:p
1469:,
1466:)
1461:1
1457:a
1453:(
1450:p
1445:(
1437:{
1432:=
1428:C
1406:C
1392:F
1384:α
1380:α
1376:α
1372:α
1368:q
1364:n
1360:α
1356:α
1352:α
1348:n
1344:F
1328:n
1324:a
1320:,
1314:,
1309:1
1305:a
1294:q
1290:n
1286:p
1272:q
1252:F
1242:k
1228:p
1208:k
1176:1
1170:q
1167:=
1164:n
1144:q
1141:=
1138:n
1116:n
1113:k
1108:=
1105:R
1078:q
1058:q
1038:F
1015:q
1009:n
1003:k
993:,
980:k
957:n
934:q
678:K
674:N
670:N
666:K
659:K
655:N
651:N
647:K
643:N
392:n
388:k
353:t
349:b
343:b
334:t
324:t
322:⌊
314:t
310:k
306:n
302:t
215:e
208:t
201:v
156:q
142:q
138:n
132:p
128:n
124:p
120:q
107:k
103:n
91:k
79:n
20:)
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