867:
639:
862:{\displaystyle {\begin{aligned}\mathbb {P} (x{\mbox{ received}}\mid y{\mbox{ sent}})&{}={\frac {\mathbb {P} (x{\mbox{ received}},y{\mbox{ sent}})}{\mathbb {P} (y{\mbox{ sent}})}}\\&{}=\mathbb {P} (y{\mbox{ sent}}\mid x{\mbox{ received}})\cdot {\frac {\mathbb {P} (x{\mbox{ received}})}{\mathbb {P} (y{\mbox{ sent}})}}.\end{aligned}}}
1576:
392:
Each codeword does not have an expected possibility: there may be more than one codeword with an equal likelihood of mutating into the received message. In such a case, the sender and receiver(s) must agree ahead of time on a decoding convention. Popular conventions include:
1402:
2540:
1088:
1017:
560:
339:
1841:
3044:
1407:
644:
908:
1266:
1571:{\displaystyle {\begin{aligned}\mathbb {P} (y{\mbox{ received}}\mid x{\mbox{ sent}})&{}=(1-p)^{n-d}\cdot p^{d}\\&{}=(1-p)^{n}\cdot \left({\frac {p}{1-p}}\right)^{d}\\\end{aligned}}}
1706:
966:
88:
1947:
1907:
1153:
471:
255:
2329:
2905:
172:
2114:
1315:
1391:
2612:
2767:
2742:
2697:
2373:
2204:
2143:
2055:
1183:
503:
285:
207:
2240:
1979:
2796:
2175:
2029:
138:
2469:
2396:
2968:
2945:
2925:
2856:
2836:
2816:
2717:
2672:
2652:
2632:
2586:
2446:
2416:
1999:
1864:
1789:
1769:
1746:
1726:
1624:
1335:
1309:
1289:
1037:
928:
627:
604:
584:
382:
362:
112:
1639:. They may be unreasonable for other media, such as a DVD, where a single scratch on the disk can cause an error in many neighbouring symbols or codewords.
3390:
2565:-probabilistic methods all based on the observation that it is easier to guess enough error-free positions, than it is to guess all the error-positions.
2477:
1042:
971:
514:
293:
1103:
The maximum likelihood decoding algorithm is an instance of the "marginalize a product function" problem which is solved by applying the
1797:
2973:
3203:
3149:
629:
was sent. If all codewords are equally likely to be sent then this scheme is equivalent to ideal observer decoding. In fact, by
3147:
Feldman, Jon; Wainwright, Martin J.; Karger, David R. (March 2005). "Using Linear
Programming to Decode Binary Linear Codes".
3435:
3404:
3376:
3316:
3283:
35:. There have been many common methods of mapping messages to codewords. These are often used to recover messages sent over a
3071:) is a method for converting the weak analog signal from the head of a magnetic disk or tape drive into a digital signal.
1629:
Errors are independent events – an error at one position in the message does not affect other positions.
875:
410:
3246:
1195:
1671:
933:
53:
1866:
errors are made then minimum distance decoding will still correctly decode the incorrectly transmitted codeword).
3122:
1912:
1872:
1118:
436:
220:
3423:
2248:
1104:
143:
3238:
2063:
506:
413:, mark the ambiguous bits of the codeword as erasures and hope that the outer code disambiguates them
28:
3163:
3086:
2339:
1636:
1600:. Minimum distance decoding is a reasonable decoding method when the following conditions are met:
400:
40:
3085:
A Viterbi decoder uses the
Viterbi algorithm for decoding a bitstream that has been encoded using
3368:
2861:
1351:
607:
2947:
contained no errors, and hence equalled the positions of the sent codeword, then we may decode.
3454:
3230:
3158:
2591:
3195:
2345:
2183:
2122:
2034:
1162:
482:
264:
177:
3396:
2213:
1952:
2772:
2151:
8:
3117:
2562:
2004:
1749:
1097:
1093:
As with ideal observer decoding, a convention must be agreed to for non-unique decoding.
406:
Choose any random codeword from the set of most likely codewords which is nearer to that.
117:
2747:
2722:
2677:
2451:
2378:
1642:
As with other decoding methods, a convention must be agreed to for non-unique decoding.
3416:
3361:
3322:
3176:
3097:
2953:
2930:
2910:
2841:
2821:
2801:
2702:
2657:
2637:
2617:
2571:
2431:
2401:
1984:
1849:
1774:
1754:
1731:
1711:
1609:
1320:
1294:
1274:
1022:
913:
612:
589:
569:
474:
428:
367:
347:
97:
2970:
errors occurred, the probability of such a fortunate selection of columns is given by
3431:
3400:
3372:
3312:
3279:
3242:
3326:
3304:
3271:
3212:
3180:
3168:
3090:
1186:
3427:
3267:
3127:
3080:
2422:
2145:. Syndrome decoding takes advantage of the property of the parity matrix that:
1597:
3448:
3050:
2551:
630:
36:
20:
3172:
2535:{\displaystyle {\begin{matrix}\sum _{i=0}^{t}{\binom {n}{i}}\\\end{matrix}}}
1665:
using a reduced lookup table. This is allowed by the linearity of the code.
3308:
3108:
Optimal decision decoding algorithm (ODDA) for an asymmetric TWRC system.
2421:
Note that this is already of significantly less complexity than that of a
3342:
Optimal decision decoding algorithm (ODDA) for an asymmetric TWRC system;
3262:
Stern, Jacques (1989). "A method for finding codewords of small weight".
2448:
errors were made during transmission, the receiver can look up the value
1654:
91:
1981:
is received. Ordinary minimum distance decoding would lookup the vector
968:
is constant as all codewords are equally likely to be sent. Therefore,
3386:
3303:. Lecture Notes in Computer Science. Vol. 1514. pp. 187–199.
3275:
3216:
1661:, i.e. one on which errors are made. In essence, syndrome decoding is
1083:{\displaystyle \mathbb {P} (y{\mbox{ sent}}\mid x{\mbox{ received}})}
1012:{\displaystyle \mathbb {P} (x{\mbox{ received}}\mid y{\mbox{ sent}})}
555:{\displaystyle \mathbb {P} (x{\mbox{ received}}\mid y{\mbox{ sent}})}
334:{\displaystyle \mathbb {P} (y{\mbox{ sent}}\mid x{\mbox{ received}})}
2769:
be the sub-vector for the corresponding positions of any codeword
1626:
that an error occurs is independent of the position of the symbol.
1096:
The maximum likelihood decoding problem can also be modeled as an
3049:
This method has been improved in various ways, e.g. by Stern and
2031:
for the nearest match - i.e. an element (not necessarily unique)
3103:
1836:{\displaystyle t=\left\lfloor {\frac {d-1}{2}}\right\rfloor }
1635:
These assumptions may be reasonable for transmissions over a
3344:, Universal Journal of Electrical and Electronic Engineering
3093:
is used as a metric for hard decision
Viterbi decoders. The
3039:{\displaystyle \textstyle {\binom {n-t}{k}}/{\binom {n}{k}}}
3367:. Oxford Applied Mathematics and Computing Science Series.
3223:
3068:
3062:
32:
3146:
3229:
3056:
2977:
2482:
1435:
1422:
1071:
1058:
1000:
987:
949:
891:
840:
817:
790:
777:
741:
718:
705:
672:
659:
543:
530:
399:
Request that the codeword be resent –
322:
309:
3395:. Wiley-Interscience Series in Discrete Mathematics.
2976:
2956:
2933:
2913:
2864:
2844:
2824:
2804:
2775:
2750:
2725:
2705:
2680:
2660:
2640:
2620:
2594:
2574:
2480:
2454:
2434:
2404:
2381:
2348:
2251:
2216:
2186:
2154:
2125:
2066:
2037:
2007:
1987:
1955:
1915:
1875:
1852:
1800:
1777:
1757:
1734:
1714:
1674:
1612:
1405:
1354:
1323:
1297:
1277:
1198:
1165:
1121:
1045:
1025:
974:
936:
916:
878:
642:
615:
592:
572:
517:
485:
439:
370:
350:
296:
267:
223:
180:
146:
120:
100:
56:
27:
is the process of translating received messages into
3392:
Introduction to the theory of error-correcting codes
3266:. Lecture Notes in Computer Science. Vol. 388.
3194:
Aji, Srinivas M.; McEliece, Robert J. (March 2000).
3415:
3360:
3038:
2962:
2939:
2919:
2899:
2850:
2830:
2810:
2790:
2761:
2736:
2711:
2691:
2666:
2646:
2626:
2606:
2580:
2534:
2463:
2440:
2410:
2390:
2367:
2323:
2234:
2198:
2169:
2137:
2108:
2049:
2023:
1993:
1973:
1941:
1901:
1858:
1846:errors made by the channel (since if no more than
1835:
1783:
1763:
1740:
1720:
1700:
1618:
1585:is less than one half) is maximised by minimising
1570:
1385:
1329:
1303:
1283:
1260:
1177:
1147:
1082:
1031:
1011:
960:
922:
902:
861:
621:
598:
578:
554:
497:
465:
376:
364:that is most likely to be received as the message
356:
333:
279:
249:
201:
166:
132:
106:
82:
2342:, one has to look-up a precomputed table of size
903:{\displaystyle \mathbb {P} (x{\mbox{ received}})}
3446:
3187:
3100:is used as a metric for soft decision decoders.
2744:will have full rank, which means that if we let
2428:However, under the assumption that no more than
2335:
3339:
1909:is sent over the channel and the error pattern
1261:{\displaystyle d(x,y)=\#\{i:x_{i}\not =y_{i}\}}
422:
344:For example, a person can choose the codeword
3029:
3016:
3002:
2981:
2522:
2509:
1701:{\displaystyle C\subset \mathbb {F} _{2}^{n}}
1596:. It can be assisted or automated by using a
961:{\displaystyle \mathbb {P} (y{\mbox{ sent}})}
83:{\displaystyle C\subset \mathbb {F} _{2}^{n}}
1255:
1223:
1110:
3193:
3140:
2556:
1653:is a highly efficient method of decoding a
1592:Minimum distance decoding is also known as
1314:Note that if the probability of error on a
1019:is maximised as a function of the variable
3104:Optimal decision decoding algorithm (ODDA)
212:
3255:
3162:
1942:{\displaystyle e\in \mathbb {F} _{2}^{n}}
1924:
1902:{\displaystyle x\in \mathbb {F} _{2}^{n}}
1884:
1683:
1411:
1382:
1148:{\displaystyle x\in \mathbb {F} _{2}^{n}}
1130:
1047:
976:
938:
880:
829:
806:
766:
730:
694:
648:
519:
466:{\displaystyle x\in \mathbb {F} _{2}^{n}}
448:
298:
250:{\displaystyle x\in \mathbb {F} _{2}^{n}}
232:
149:
65:
3413:
3292:
2568:The simplest form is due to Prange: Let
287:. The process results in this solution:
209:is the distance between those elements.
3299:Ohta, Kazuo; Pei, Dingyi, eds. (1998).
3298:
3204:IEEE Transactions on Information Theory
3150:IEEE Transactions on Information Theory
2324:{\displaystyle Hz=H(x+e)=Hx+He=0+He=He}
416:Report a decoding failure to the system
387:
3447:
3385:
3301:Advances in Cryptology — ASIACRYPT'98
3261:
3067:Partial response maximum likelihood (
2907:. Hence, if we were lucky that these
1337:is strictly less than one half, then
1090:is maximised, and the claim follows.
3358:
1645:
586:that maximizes the probability that
167:{\displaystyle \mathbb {F} _{2}^{n}}
3089:based on a convolutional code. The
3057:Partial response maximum likelihood
2471:in a further reduced table of size
13:
3426:(GTM). Vol. 86 (2 ed.).
3352:
3196:"The Generalized Distributive Law"
3074:
3020:
2985:
2513:
1220:
14:
3466:
2109:{\displaystyle d(c,z)\leq d(y,z)}
2545:
1291:that is as close as possible to
3363:A first course in coding theory
2927:positions of the received word
2699:the corresponding submatrix of
1791:is capable of correcting up to
3333:
3264:Coding Theory and Applications
3123:Error detection and correction
2719:. With reasonable probability
2276:
2264:
2103:
2091:
2082:
2070:
2017:
2009:
1519:
1506:
1466:
1453:
1441:
1415:
1370:
1358:
1214:
1202:
1077:
1051:
1006:
980:
955:
942:
897:
884:
846:
833:
823:
810:
796:
770:
747:
734:
724:
698:
678:
652:
549:
523:
328:
302:
196:
184:
1:
3424:Graduate Texts in Mathematics
3418:Introduction to Coding Theory
3414:van Lint, Jacobus H. (1992).
3133:
217:One may be given the message
16:Algorithms to decode messages
1869:Now suppose that a codeword
1105:generalized distributive law
7:
3111:
2900:{\displaystyle m=c'G'^{-1}}
1708:is a linear code of length
1386:{\displaystyle d(x,y)=d,\,}
1343:maximum likelihood decoding
1316:discrete memoryless channel
423:Maximum likelihood decoding
46:
10:
3471:
3239:Cambridge University Press
3078:
3060:
2634:used for encoding. Select
2549:
1594:nearest neighbour decoding
1115:Given a received codeword
426:
3233:; Rosenbaum, Ute (1998).
2674:at random, and denote by
2607:{\displaystyle k\times n}
1663:minimum distance decoding
1339:minimum distance decoding
1271:i.e. choose the codeword
1157:minimum distance decoding
1111:Minimum distance decoding
3340:Siamack Ghadimi (2020),
3087:forward error correction
2557:Information set decoding
2340:binary symmetric channel
1637:binary symmetric channel
433:Given a received vector
401:automatic repeat-request
41:binary symmetric channel
3369:Oxford University Press
3231:Beutelspacher, Albrecht
3173:10.1109/TIT.2004.842696
2423:standard array decoding
2368:{\displaystyle 2^{n-k}}
261:generates the codeword
259:ideal observer decoding
213:Ideal observer decoding
3359:Hill, Raymond (1986).
3040:
2964:
2941:
2921:
2901:
2852:
2832:
2812:
2792:
2763:
2738:
2713:
2693:
2668:
2648:
2628:
2608:
2582:
2536:
2505:
2465:
2442:
2412:
2392:
2369:
2325:
2236:
2200:
2199:{\displaystyle x\in C}
2171:
2139:
2138:{\displaystyle y\in C}
2110:
2051:
2050:{\displaystyle c\in C}
2025:
1995:
1975:
1943:
1903:
1860:
1837:
1785:
1765:
1742:
1722:
1702:
1620:
1572:
1387:
1331:
1305:
1285:
1262:
1179:
1178:{\displaystyle y\in C}
1149:
1084:
1033:
1013:
962:
924:
904:
863:
623:
600:
580:
566:that is, the codeword
556:
499:
498:{\displaystyle y\in C}
467:
378:
358:
335:
281:
280:{\displaystyle y\in C}
251:
203:
202:{\displaystyle d(x,y)}
168:
134:
108:
84:
3397:John Wiley & Sons
3309:10.1007/3-540-49649-1
3041:
2965:
2942:
2922:
2902:
2853:
2833:
2813:
2793:
2764:
2739:
2714:
2694:
2669:
2649:
2629:
2609:
2583:
2537:
2485:
2466:
2443:
2413:
2393:
2370:
2326:
2237:
2235:{\displaystyle z=x+e}
2201:
2172:
2140:
2111:
2052:
2026:
1996:
1976:
1974:{\displaystyle z=x+e}
1944:
1904:
1861:
1838:
1786:
1766:
1743:
1728:and minimum distance
1723:
1703:
1621:
1573:
1388:
1332:
1306:
1286:
1263:
1180:
1150:
1085:
1034:
1014:
963:
925:
905:
864:
624:
601:
581:
557:
500:
468:
427:Further information:
379:
359:
336:
282:
252:
204:
169:
140:shall be elements of
135:
109:
85:
3270:. pp. 106–113.
2974:
2954:
2931:
2911:
2862:
2842:
2822:
2802:
2791:{\displaystyle c=mG}
2773:
2748:
2723:
2703:
2678:
2658:
2638:
2618:
2614:generator matrix of
2592:
2572:
2561:This is a family of
2478:
2452:
2432:
2402:
2379:
2346:
2249:
2214:
2184:
2170:{\displaystyle Hx=0}
2152:
2123:
2064:
2035:
2005:
1985:
1953:
1913:
1873:
1850:
1798:
1775:
1755:
1732:
1712:
1672:
1610:
1403:
1352:
1321:
1295:
1275:
1196:
1163:
1119:
1043:
1023:
972:
934:
930:is restructured and
914:
876:
640:
613:
590:
570:
515:
483:
437:
411:another code follows
388:Decoding conventions
384:after transmission.
368:
348:
294:
265:
221:
178:
144:
118:
98:
54:
3235:Projective Geometry
2024:{\displaystyle |C|}
2001:in a table of size
1938:
1898:
1750:parity-check matrix
1697:
1144:
1098:integer programming
462:
246:
163:
133:{\displaystyle x,y}
79:
3276:10.1007/BFb0019850
3098:Euclidean distance
3036:
3035:
2960:
2937:
2917:
2897:
2848:
2828:
2808:
2788:
2762:{\displaystyle c'}
2759:
2737:{\displaystyle G'}
2734:
2709:
2692:{\displaystyle G'}
2689:
2664:
2644:
2624:
2604:
2578:
2532:
2530:
2464:{\displaystyle He}
2461:
2438:
2408:
2391:{\displaystyle He}
2388:
2365:
2321:
2242:is defined to be:
2232:
2196:
2167:
2135:
2106:
2047:
2021:
1991:
1971:
1939:
1922:
1899:
1882:
1856:
1833:
1781:
1761:
1738:
1718:
1698:
1681:
1616:
1568:
1566:
1439:
1426:
1383:
1327:
1301:
1281:
1258:
1175:
1145:
1128:
1080:
1075:
1062:
1029:
1009:
1004:
991:
958:
953:
920:
900:
895:
859:
857:
844:
821:
794:
781:
745:
722:
709:
676:
663:
619:
596:
576:
552:
547:
534:
495:
475:maximum likelihood
463:
446:
429:Maximum likelihood
374:
354:
331:
326:
313:
277:
247:
230:
199:
164:
147:
130:
104:
80:
63:
3437:978-3-540-54894-2
3406:978-0-471-08684-0
3378:978-0-19-853803-5
3318:978-3-540-65109-3
3285:978-3-540-51643-9
3217:10.1109/18.825794
3027:
3000:
2963:{\displaystyle t}
2940:{\displaystyle y}
2920:{\displaystyle k}
2851:{\displaystyle m}
2838:, we can recover
2831:{\displaystyle m}
2811:{\displaystyle C}
2712:{\displaystyle G}
2667:{\displaystyle G}
2647:{\displaystyle k}
2627:{\displaystyle C}
2581:{\displaystyle G}
2520:
2441:{\displaystyle t}
2411:{\displaystyle e}
1994:{\displaystyle z}
1859:{\displaystyle t}
1827:
1784:{\displaystyle C}
1764:{\displaystyle H}
1741:{\displaystyle d}
1721:{\displaystyle n}
1651:Syndrome decoding
1646:Syndrome decoding
1619:{\displaystyle p}
1552:
1438:
1425:
1341:is equivalent to
1330:{\displaystyle p}
1304:{\displaystyle x}
1284:{\displaystyle y}
1159:picks a codeword
1074:
1061:
1032:{\displaystyle y}
1003:
990:
952:
923:{\displaystyle x}
894:
850:
843:
820:
793:
780:
751:
744:
721:
708:
675:
662:
622:{\displaystyle y}
599:{\displaystyle x}
579:{\displaystyle y}
546:
533:
479:picks a codeword
377:{\displaystyle x}
357:{\displaystyle y}
325:
312:
107:{\displaystyle n}
3462:
3441:
3421:
3410:
3382:
3366:
3346:
3345:
3337:
3331:
3330:
3296:
3290:
3289:
3259:
3253:
3252:
3227:
3221:
3220:
3200:
3191:
3185:
3184:
3166:
3144:
3118:Don't care alarm
3091:Hamming distance
3045:
3043:
3042:
3037:
3034:
3033:
3032:
3019:
3012:
3007:
3006:
3005:
2996:
2984:
2969:
2967:
2966:
2961:
2946:
2944:
2943:
2938:
2926:
2924:
2923:
2918:
2906:
2904:
2903:
2898:
2896:
2895:
2894:
2878:
2857:
2855:
2854:
2849:
2837:
2835:
2834:
2829:
2817:
2815:
2814:
2809:
2797:
2795:
2794:
2789:
2768:
2766:
2765:
2760:
2758:
2743:
2741:
2740:
2735:
2733:
2718:
2716:
2715:
2710:
2698:
2696:
2695:
2690:
2688:
2673:
2671:
2670:
2665:
2653:
2651:
2650:
2645:
2633:
2631:
2630:
2625:
2613:
2611:
2610:
2605:
2587:
2585:
2584:
2579:
2541:
2539:
2538:
2533:
2531:
2527:
2526:
2525:
2512:
2504:
2499:
2470:
2468:
2467:
2462:
2447:
2445:
2444:
2439:
2417:
2415:
2414:
2409:
2397:
2395:
2394:
2389:
2374:
2372:
2371:
2366:
2364:
2363:
2330:
2328:
2327:
2322:
2241:
2239:
2238:
2233:
2210:of the received
2205:
2203:
2202:
2197:
2176:
2174:
2173:
2168:
2144:
2142:
2141:
2136:
2115:
2113:
2112:
2107:
2056:
2054:
2053:
2048:
2030:
2028:
2027:
2022:
2020:
2012:
2000:
1998:
1997:
1992:
1980:
1978:
1977:
1972:
1948:
1946:
1945:
1940:
1937:
1932:
1927:
1908:
1906:
1905:
1900:
1897:
1892:
1887:
1865:
1863:
1862:
1857:
1842:
1840:
1839:
1834:
1832:
1828:
1823:
1812:
1790:
1788:
1787:
1782:
1770:
1768:
1767:
1762:
1747:
1745:
1744:
1739:
1727:
1725:
1724:
1719:
1707:
1705:
1704:
1699:
1696:
1691:
1686:
1625:
1623:
1622:
1617:
1606:The probability
1577:
1575:
1574:
1569:
1567:
1563:
1562:
1557:
1553:
1551:
1537:
1527:
1526:
1502:
1497:
1493:
1492:
1480:
1479:
1449:
1440:
1436:
1427:
1423:
1414:
1392:
1390:
1389:
1384:
1336:
1334:
1333:
1328:
1310:
1308:
1307:
1302:
1290:
1288:
1287:
1282:
1267:
1265:
1264:
1259:
1254:
1253:
1241:
1240:
1187:Hamming distance
1185:to minimise the
1184:
1182:
1181:
1176:
1154:
1152:
1151:
1146:
1143:
1138:
1133:
1089:
1087:
1086:
1081:
1076:
1072:
1063:
1059:
1050:
1038:
1036:
1035:
1030:
1018:
1016:
1015:
1010:
1005:
1001:
992:
988:
979:
967:
965:
964:
959:
954:
950:
941:
929:
927:
926:
921:
909:
907:
906:
901:
896:
892:
883:
868:
866:
865:
860:
858:
851:
849:
845:
841:
832:
826:
822:
818:
809:
803:
795:
791:
782:
778:
769:
761:
756:
752:
750:
746:
742:
733:
727:
723:
719:
710:
706:
697:
691:
686:
677:
673:
664:
660:
651:
628:
626:
625:
620:
605:
603:
602:
597:
585:
583:
582:
577:
561:
559:
558:
553:
548:
544:
535:
531:
522:
504:
502:
501:
496:
472:
470:
469:
464:
461:
456:
451:
383:
381:
380:
375:
363:
361:
360:
355:
340:
338:
337:
332:
327:
323:
314:
310:
301:
286:
284:
283:
278:
256:
254:
253:
248:
245:
240:
235:
208:
206:
205:
200:
173:
171:
170:
165:
162:
157:
152:
139:
137:
136:
131:
113:
111:
110:
105:
94:with the length
90:is considered a
89:
87:
86:
81:
78:
73:
68:
3470:
3469:
3465:
3464:
3463:
3461:
3460:
3459:
3445:
3444:
3438:
3428:Springer-Verlag
3407:
3379:
3355:
3353:Further reading
3350:
3349:
3338:
3334:
3319:
3297:
3293:
3286:
3268:Springer-Verlag
3260:
3256:
3249:
3241:. p. 190.
3228:
3224:
3198:
3192:
3188:
3164:10.1.1.111.6585
3145:
3141:
3136:
3128:Forbidden input
3114:
3106:
3083:
3081:Viterbi decoder
3077:
3075:Viterbi decoder
3065:
3059:
3028:
3015:
3014:
3013:
3008:
3001:
2986:
2980:
2979:
2978:
2975:
2972:
2971:
2955:
2952:
2951:
2932:
2929:
2928:
2912:
2909:
2908:
2887:
2883:
2879:
2871:
2863:
2860:
2859:
2843:
2840:
2839:
2823:
2820:
2819:
2803:
2800:
2799:
2774:
2771:
2770:
2751:
2749:
2746:
2745:
2726:
2724:
2721:
2720:
2704:
2701:
2700:
2681:
2679:
2676:
2675:
2659:
2656:
2655:
2639:
2636:
2635:
2619:
2616:
2615:
2593:
2590:
2589:
2573:
2570:
2569:
2559:
2554:
2548:
2529:
2528:
2521:
2508:
2507:
2506:
2500:
2489:
2481:
2479:
2476:
2475:
2453:
2450:
2449:
2433:
2430:
2429:
2403:
2400:
2399:
2380:
2377:
2376:
2353:
2349:
2347:
2344:
2343:
2250:
2247:
2246:
2215:
2212:
2211:
2185:
2182:
2181:
2153:
2150:
2149:
2124:
2121:
2120:
2065:
2062:
2061:
2036:
2033:
2032:
2016:
2008:
2006:
2003:
2002:
1986:
1983:
1982:
1954:
1951:
1950:
1933:
1928:
1923:
1914:
1911:
1910:
1893:
1888:
1883:
1874:
1871:
1870:
1851:
1848:
1847:
1813:
1811:
1807:
1799:
1796:
1795:
1776:
1773:
1772:
1771:. Then clearly
1756:
1753:
1752:
1733:
1730:
1729:
1713:
1710:
1709:
1692:
1687:
1682:
1673:
1670:
1669:
1648:
1611:
1608:
1607:
1565:
1564:
1558:
1541:
1536:
1532:
1531:
1522:
1518:
1501:
1495:
1494:
1488:
1484:
1469:
1465:
1448:
1444:
1434:
1421:
1410:
1406:
1404:
1401:
1400:
1353:
1350:
1349:
1322:
1319:
1318:
1296:
1293:
1292:
1276:
1273:
1272:
1249:
1245:
1236:
1232:
1197:
1194:
1193:
1164:
1161:
1160:
1139:
1134:
1129:
1120:
1117:
1116:
1113:
1070:
1057:
1046:
1044:
1041:
1040:
1039:precisely when
1024:
1021:
1020:
999:
986:
975:
973:
970:
969:
948:
937:
935:
932:
931:
915:
912:
911:
890:
879:
877:
874:
873:
856:
855:
839:
828:
827:
816:
805:
804:
802:
789:
776:
765:
760:
754:
753:
740:
729:
728:
717:
704:
693:
692:
690:
685:
681:
671:
658:
647:
643:
641:
638:
637:
614:
611:
610:
591:
588:
587:
571:
568:
567:
542:
529:
518:
516:
513:
512:
484:
481:
480:
457:
452:
447:
438:
435:
434:
431:
425:
390:
369:
366:
365:
349:
346:
345:
321:
308:
297:
295:
292:
291:
266:
263:
262:
241:
236:
231:
222:
219:
218:
215:
179:
176:
175:
158:
153:
148:
145:
142:
141:
119:
116:
115:
99:
96:
95:
74:
69:
64:
55:
52:
51:
49:
17:
12:
11:
5:
3468:
3458:
3457:
3443:
3442:
3436:
3411:
3405:
3383:
3377:
3354:
3351:
3348:
3347:
3332:
3317:
3291:
3284:
3254:
3247:
3222:
3211:(2): 325–343.
3186:
3157:(3): 954–972.
3138:
3137:
3135:
3132:
3131:
3130:
3125:
3120:
3113:
3110:
3105:
3102:
3079:Main article:
3076:
3073:
3061:Main article:
3058:
3055:
3053:and Sendrier.
3031:
3026:
3023:
3018:
3011:
3004:
2999:
2995:
2992:
2989:
2983:
2959:
2936:
2916:
2893:
2890:
2886:
2882:
2877:
2874:
2870:
2867:
2847:
2827:
2818:for a message
2807:
2787:
2784:
2781:
2778:
2757:
2754:
2732:
2729:
2708:
2687:
2684:
2663:
2643:
2623:
2603:
2600:
2597:
2577:
2558:
2555:
2550:Main article:
2547:
2544:
2543:
2542:
2524:
2519:
2516:
2511:
2503:
2498:
2495:
2492:
2488:
2484:
2483:
2460:
2457:
2437:
2407:
2387:
2384:
2362:
2359:
2356:
2352:
2332:
2331:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2231:
2228:
2225:
2222:
2219:
2195:
2192:
2189:
2178:
2177:
2166:
2163:
2160:
2157:
2134:
2131:
2128:
2117:
2116:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2046:
2043:
2040:
2019:
2015:
2011:
1990:
1970:
1967:
1964:
1961:
1958:
1936:
1931:
1926:
1921:
1918:
1896:
1891:
1886:
1881:
1878:
1855:
1844:
1843:
1831:
1826:
1822:
1819:
1816:
1810:
1806:
1803:
1780:
1760:
1737:
1717:
1695:
1690:
1685:
1680:
1677:
1647:
1644:
1633:
1632:
1631:
1630:
1627:
1615:
1598:standard array
1579:
1578:
1561:
1556:
1550:
1547:
1544:
1540:
1535:
1530:
1525:
1521:
1517:
1514:
1511:
1508:
1505:
1500:
1498:
1496:
1491:
1487:
1483:
1478:
1475:
1472:
1468:
1464:
1461:
1458:
1455:
1452:
1447:
1445:
1443:
1433:
1430:
1424: received
1420:
1417:
1413:
1409:
1408:
1394:
1393:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1326:
1300:
1280:
1269:
1268:
1257:
1252:
1248:
1244:
1239:
1235:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1174:
1171:
1168:
1142:
1137:
1132:
1127:
1124:
1112:
1109:
1079:
1073: received
1069:
1066:
1056:
1053:
1049:
1028:
1008:
998:
995:
989: received
985:
982:
978:
957:
947:
944:
940:
919:
899:
893: received
889:
886:
882:
870:
869:
854:
848:
838:
835:
831:
825:
819: received
815:
812:
808:
801:
798:
792: received
788:
785:
775:
772:
768:
764:
759:
757:
755:
749:
739:
736:
732:
726:
716:
713:
707: received
703:
700:
696:
689:
684:
682:
680:
670:
667:
661: received
657:
654:
650:
646:
645:
618:
606:was received,
595:
575:
564:
563:
551:
541:
538:
532: received
528:
525:
521:
494:
491:
488:
460:
455:
450:
445:
442:
424:
421:
420:
419:
418:
417:
414:
407:
404:
389:
386:
373:
353:
342:
341:
330:
324: received
320:
317:
307:
304:
300:
276:
273:
270:
244:
239:
234:
229:
226:
214:
211:
198:
195:
192:
189:
186:
183:
161:
156:
151:
129:
126:
123:
103:
77:
72:
67:
62:
59:
48:
45:
15:
9:
6:
4:
3:
2:
3467:
3456:
3455:Coding theory
3453:
3452:
3450:
3439:
3433:
3429:
3425:
3420:
3419:
3412:
3408:
3402:
3398:
3394:
3393:
3388:
3384:
3380:
3374:
3370:
3365:
3364:
3357:
3356:
3343:
3336:
3328:
3324:
3320:
3314:
3310:
3306:
3302:
3295:
3287:
3281:
3277:
3273:
3269:
3265:
3258:
3250:
3248:0-521-48277-1
3244:
3240:
3236:
3232:
3226:
3218:
3214:
3210:
3206:
3205:
3197:
3190:
3182:
3178:
3174:
3170:
3165:
3160:
3156:
3152:
3151:
3143:
3139:
3129:
3126:
3124:
3121:
3119:
3116:
3115:
3109:
3101:
3099:
3096:
3092:
3088:
3082:
3072:
3070:
3064:
3054:
3052:
3047:
3024:
3021:
3009:
2997:
2993:
2990:
2987:
2957:
2948:
2934:
2914:
2891:
2888:
2884:
2880:
2875:
2872:
2868:
2865:
2845:
2825:
2805:
2785:
2782:
2779:
2776:
2755:
2752:
2730:
2727:
2706:
2685:
2682:
2661:
2641:
2621:
2601:
2598:
2595:
2575:
2566:
2564:
2553:
2552:List decoding
2546:List decoding
2517:
2514:
2501:
2496:
2493:
2490:
2486:
2474:
2473:
2472:
2458:
2455:
2435:
2426:
2424:
2419:
2405:
2385:
2382:
2360:
2357:
2354:
2350:
2341:
2337:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2273:
2270:
2267:
2261:
2258:
2255:
2252:
2245:
2244:
2243:
2229:
2226:
2223:
2220:
2217:
2209:
2193:
2190:
2187:
2164:
2161:
2158:
2155:
2148:
2147:
2146:
2132:
2129:
2126:
2100:
2097:
2094:
2088:
2085:
2079:
2076:
2073:
2067:
2060:
2059:
2058:
2044:
2041:
2038:
2013:
1988:
1968:
1965:
1962:
1959:
1956:
1949:occurs. Then
1934:
1929:
1919:
1916:
1894:
1889:
1879:
1876:
1867:
1853:
1829:
1824:
1820:
1817:
1814:
1808:
1804:
1801:
1794:
1793:
1792:
1778:
1758:
1751:
1735:
1715:
1693:
1688:
1678:
1675:
1668:Suppose that
1666:
1664:
1660:
1659:noisy channel
1656:
1652:
1643:
1640:
1638:
1628:
1613:
1605:
1604:
1603:
1602:
1601:
1599:
1595:
1590:
1588:
1584:
1581:which (since
1559:
1554:
1548:
1545:
1542:
1538:
1533:
1528:
1523:
1515:
1512:
1509:
1503:
1499:
1489:
1485:
1481:
1476:
1473:
1470:
1462:
1459:
1456:
1450:
1446:
1431:
1428:
1418:
1399:
1398:
1397:
1379:
1376:
1373:
1367:
1364:
1361:
1355:
1348:
1347:
1346:
1344:
1340:
1324:
1317:
1312:
1298:
1278:
1250:
1246:
1242:
1237:
1233:
1229:
1226:
1217:
1211:
1208:
1205:
1199:
1192:
1191:
1190:
1188:
1172:
1169:
1166:
1158:
1140:
1135:
1125:
1122:
1108:
1106:
1101:
1099:
1094:
1091:
1067:
1064:
1054:
1026:
996:
993:
983:
945:
917:
887:
852:
836:
813:
799:
786:
783:
773:
762:
758:
737:
714:
711:
701:
687:
683:
668:
665:
655:
636:
635:
634:
632:
631:Bayes Theorem
616:
609:
593:
573:
539:
536:
526:
511:
510:
509:
508:
492:
489:
486:
478:
476:
458:
453:
443:
440:
430:
415:
412:
408:
405:
402:
398:
397:
396:
395:
394:
385:
371:
351:
318:
315:
305:
290:
289:
288:
274:
271:
268:
260:
242:
237:
227:
224:
210:
193:
190:
187:
181:
159:
154:
127:
124:
121:
101:
93:
75:
70:
60:
57:
44:
42:
38:
37:noisy channel
34:
30:
26:
22:
21:coding theory
3417:
3391:
3362:
3341:
3335:
3300:
3294:
3263:
3257:
3234:
3225:
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2654:columns of
2336:ML decoding
2334:To perform
1655:linear code
1345:, since if
92:binary code
31:of a given
3134:References
2375:, mapping
1437: sent
1060: sent
1002: sent
951: sent
842: sent
779: sent
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674: sent
608:given that
545: sent
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3159:CiteSeerX
2991:−
2889:−
2599:×
2563:Las Vegas
2487:∑
2358:−
2191:∈
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1221:#
1170:∈
1126:∈
1100:problem.
1065:∣
994:∣
800:⋅
784:∣
666:∣
537:∣
507:maximizes
490:∈
444:∈
316:∣
272:∈
228:∈
61:⊂
29:codewords
3449:Category
3389:(1982).
3327:37257901
3112:See also
3051:Canteaut
2885:′
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2756:′
2731:′
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2208:syndrome
2180:for all
2119:for all
1830:⌋
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1243:≠
477:decoding
47:Notation
25:decoding
3181:3120399
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2588:be the
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