2385:
2089:
260:
978:
2562:
514:
1495:
1433:
1175:
2775:
2491:
2017:
1647:
833:
2604:
1254:
1216:
1803:
381:
872:
2438:
1846:
2380:{\displaystyle A_{w}={\binom {n}{w}}\sum _{j=0}^{w-d}(-1)^{j}{\binom {w}{j}}(q^{w-d+1-j}-1)={\binom {n}{w}}(q-1)\sum _{j=0}^{w-d}(-1)^{j}{\binom {w-1}{j}}q^{w-d-j}.}
697:
295:
2044:
1988:
1618:
1318:
1120:
588:
1750:
1286:
727:
2696:
2674:
2653:
2624:
2462:
2084:
2064:
1956:
1913:
1886:
1866:
1774:
1717:
1693:
1670:
1586:
1556:
1536:
1453:
1398:
1378:
1093:
1069:
1049:
1029:
1005:
892:
771:
747:
671:
651:
631:
608:
561:
541:
441:
421:
401:
339:
319:
179:
159:
119:
99:
79:
59:
2746:
3090:
2961:
Bruen, A.A.; Thas, J.A.; Blokhuis, A. (1988), "On M.D.S. codes, arcs in PG(n,q), with q even, and a solution of three fundamental problems of B. Segre",
17:
184:
897:
2395:
The linear independence of the columns of a generator matrix of an MDS code permits a construction of MDS codes from objects in
729:
letters of each codeword, then all resulting codewords must still be pairwise different, since all of the original codewords in
449:
1125:
1288:. Another simple proof follows from observing that the rows of any generator matrix in standard form have weight at most
1558:, they have the greatest error correcting and detecting capabilities. There are several ways to characterize MDS codes:
3237:
3207:
3169:
3122:
3100:
3077:
3046:
2837:
1512:
2496:
779:
1221:
1180:
1258:
In the linear code case a different proof of the
Singleton bound can be obtained by observing that rank of the
3195:
3114:
1458:
1403:
2787:
3229:
2751:
2467:
1993:
1623:
38:, named after Richard Collom Singleton, is a relatively crude upper bound on the size of an arbitrary
2656:
2571:
2565:
1781:
344:
3264:
3199:
3189:
3069:
3063:
1925:
894:
was arbitrary, this bound must hold for the largest possible code with these parameters, thus:
838:
2827:
3269:
2405:
1916:
1811:
676:
265:
2970:
2022:
1961:
1720:
1591:
1291:
1098:
566:
3247:
8:
3217:
2399:
1753:
1729:
1265:
1259:
706:
2974:
3055:
2986:
2681:
2659:
2638:
2609:
2447:
2069:
2049:
1941:
1898:
1871:
1851:
1759:
1702:
1678:
1655:
1571:
1541:
1521:
1497:(minimum distance 1), codes with a single parity symbol (minimum distance 2) and their
1438:
1383:
1363:
1078:
1054:
1034:
1014:
990:
877:
756:
732:
656:
636:
616:
593:
546:
526:
426:
406:
386:
324:
304:
164:
144:
104:
84:
64:
44:
3020:
2701:
3233:
3203:
3165:
3118:
3096:
3073:
3042:
2990:
2833:
1889:
773:
from each other. Thus the size of the altered code is the same as the original code.
3243:
3140:
3015:
2978:
2441:
1696:
750:
298:
3028:
Komamiya, Y. (1953), "Application of logical mathematics to information theory",
2396:
2792:
700:
3258:
3185:
3144:
2807:
2802:
2797:
31:
1928:, an explicit formula for the complete weight distribution of an MDS code.
1356:
Linear block codes that achieve equality in the
Singleton bound are called
1072:
3059:
1008:
3006:
Joshi, D.D (1958), "A Note on Upper Bounds for
Minimum Distance Codes",
3220:; Xing, Chaoping (2001). "6. Applications to algebraic coding theory".
3086:
2982:
39:
3225:
3222:
Rational points on curves over finite fields. Theory and
Applications
2626:
whose columns are the homogeneous coordinates of these points. Then,
1498:
1518:
MDS codes are an important class of block codes since, for a fixed
3224:. London Mathematical Society Lecture Note Series. Vol. 285.
1915:
coordinate positions, there is a (minimum weight) codeword whose
1508:
In the case of binary alphabets, only trivial MDS codes exist.
2829:
Latin
Squares: New Developments in the Theory and Applications
2564:
be a set of points in this projective space represented with
1924:
The last of these characterizations permits, by using the
610:
different values, independently of the remaining letters.
3131:
Singleton, R.C. (1964), "Maximum distance q-nary codes",
383:
represents the maximum number of possible codewords in a
3198:. Vol. 86 (2nd ed.). Springer-Verlag. p.
2826:
Keedwell, A. Donald; DĂ©nes, JĂłzsef (24 January 1991).
1360:. Examples of such codes include codes that have only
1336:. Joshi notes that the result was obtained earlier by
2754:
2704:
2684:
2662:
2641:
2612:
2574:
2499:
2470:
2450:
2408:
2092:
2072:
2052:
2025:
1996:
1964:
1944:
1901:
1874:
1854:
1814:
1784:
1762:
1732:
1705:
1681:
1658:
1626:
1594:
1574:
1544:
1524:
1461:
1441:
1406:
1386:
1366:
1294:
1268:
1224:
1183:
1128:
1101:
1081:
1057:
1037:
1017:
993:
900:
880:
841:
782:
759:
735:
709:
679:
659:
639:
619:
596:
569:
549:
529:
452:
429:
409:
389:
347:
327:
307:
268:
187:
167:
147:
107:
87:
67:
47:
3092:
Introduction to the Theory of Error-Correcting Codes
255:{\displaystyle d=\min _{\{x,y\in C:x\neq y\}}d(x,y)}
590:, since each letter in such a word may take one of
2769:
2740:
2690:
2668:
2647:
2618:
2598:
2556:
2485:
2456:
2432:
2379:
2078:
2058:
2038:
2011:
1982:
1950:
1907:
1880:
1860:
1840:
1797:
1768:
1744:
1711:
1687:
1664:
1641:
1612:
1580:
1550:
1530:
1489:
1447:
1427:
1392:
1372:
1312:
1280:
1248:
1210:
1169:
1114:
1095:elements, then the maximum number of codewords is
1087:
1063:
1043:
1023:
999:
972:
886:
866:
827:
765:
741:
721:
691:
665:
645:
625:
602:
582:
555:
535:
508:
435:
415:
395:
375:
333:
313:
289:
254:
173:
153:
113:
93:
73:
53:
3054:
2960:
2912:
2346:
2325:
2255:
2242:
2190:
2177:
2122:
2109:
1868:in standard form, then every square submatrix of
973:{\displaystyle |C|\leq A_{q}(n,d)\leq q^{n-d+1}.}
3256:
3216:
195:
2557:{\displaystyle K=\{P_{1},P_{2},\dots ,P_{m}\}}
2390:
776:The newly obtained codewords each have length
3184:
2825:
2551:
2506:
1328:The usual citation given for this result is
229:
199:
1511:Examples of non-trivial MDS codes include
3130:
3019:
2757:
2473:
1999:
1629:
1467:
1415:
1329:
509:{\displaystyle A_{q}(n,d)\leq q^{n-d+1}.}
136:
3036:
3027:
2924:
2900:
2852:
1345:
1337:
130:
3150:
3030:Proc. 3rd Japan. Nat. Cong. Appl. Math.
2888:
1344:, p. 72) also notes the same regarding
14:
3257:
1490:{\displaystyle (\mathbb {F} _{q})^{n}}
1358:MDS (maximum distance separable) codes
3159:
3108:
3085:
3005:
2948:
2936:
2876:
2864:
1428:{\displaystyle x\in \mathbb {F} _{q}}
1341:
1333:
446:Then the Singleton bound states that
126:
3095:(3rd ed.), Wiley Interscience,
3065:The Theory of Error-Correcting Codes
2832:. Amsterdam: Elsevier. p. 270.
1170:{\displaystyle q^{k}\leq q^{n-d+1},}
653:-ary block code of minimum distance
3153:Elements of algebraic coding theory
2046:denotes the number of codewords in
24:
3178:
3117:, vol. 134, Springer-Verlag,
3037:Ling, San; Xing, Chaoping (2004),
2329:
2246:
2181:
2113:
25:
3281:
1122:and the Singleton bound implies:
523:First observe that the number of
2770:{\displaystyle \mathbb {F} _{q}}
2486:{\displaystyle \mathbb {F} _{q}}
2012:{\displaystyle \mathbb {F} _{q}}
1649:. The following are equivalent:
1642:{\displaystyle \mathbb {F} _{q}}
2954:
2942:
1455:), codes that use the whole of
1435:, having thus minimum distance
1332:, but it was proven earlier by
982:
835:and thus, there can be at most
703:the code by deleting the first
18:Maximum distance separable code
3041:, Cambridge University Press,
3039:Coding Theory / A First Course
2930:
2918:
2906:
2894:
2882:
2870:
2858:
2846:
2819:
2735:
2705:
2698:is the generator matrix of an
2587:
2575:
2427:
2415:
2313:
2303:
2273:
2261:
2233:
2196:
2165:
2155:
1835:
1828:
1821:
1478:
1462:
939:
927:
910:
902:
828:{\displaystyle n-(d-1)=n-d+1,}
801:
789:
475:
463:
370:
358:
284:
272:
249:
237:
141:The minimum distance of a set
13:
1:
3191:Introduction to Coding Theory
3111:Coding and Information Theory
3021:10.1016/S0019-9958(58)80006-6
2999:
2913:MacWilliams & Sloane 1977
2599:{\displaystyle (N+1)\times m}
1919:is precisely these positions.
1515:and their extended versions.
1351:
1340:using a more complex proof.
1249:{\displaystyle d\leq n-k+1.}
1218:which is usually written as
1211:{\displaystyle k\leq n-d+1,}
27:Upper bound in coding theory
7:
3164:, Oxford University Press,
2781:
2391:Arcs in projective geometry
10:
3286:
3230:Cambridge University Press
3068:, North-Holland, pp.
1848:is a generator matrix for
1798:{\displaystyle C^{\perp }}
1323:
423:and minimum distance
403:-ary block code of length
376:{\displaystyle A_{q}(n,d)}
121:. It is also known as the
2444:of (geometric) dimension
1776:are linearly independent.
1501:. These are often called
867:{\displaystyle q^{n-d+1}}
673:. Clearly, all codewords
3145:10.1109/TIT.1964.1053661
2813:
518:
3160:Welsh, Dominic (1988),
3151:Vermani, L. R. (1996),
3133:IEEE Trans. Inf. Theory
3008:Information and Control
2939:, p. 237, Theorem 5.3.7
2788:Gilbert–Varshamov bound
2566:homogeneous coordinates
2433:{\displaystyle PG(N,q)}
1841:{\displaystyle G=(I|A)}
161:of codewords of length
3162:Codes and Cryptography
3109:Roman, Steven (1992),
2771:
2742:
2692:
2670:
2649:
2620:
2600:
2558:
2487:
2464:over the finite field
2458:
2434:
2381:
2302:
2154:
2080:
2060:
2040:
2013:
1984:
1952:
1926:MacWilliams identities
1909:
1882:
1862:
1842:
1799:
1770:
1746:
1713:
1689:
1666:
1643:
1614:
1582:
1552:
1532:
1491:
1449:
1429:
1394:
1374:
1314:
1282:
1250:
1212:
1171:
1116:
1089:
1065:
1045:
1025:
1001:
974:
888:
868:
829:
767:
743:
723:
693:
692:{\displaystyle c\in C}
667:
647:
627:
604:
584:
557:
537:
510:
437:
417:
397:
377:
335:
315:
291:
290:{\displaystyle d(x,y)}
256:
175:
155:
137:Statement of the bound
115:
95:
75:
55:
2772:
2743:
2693:
2671:
2650:
2621:
2601:
2559:
2488:
2459:
2435:
2382:
2276:
2128:
2081:
2061:
2041:
2039:{\displaystyle A_{w}}
2014:
1985:
1983:{\displaystyle n,k,d}
1953:
1910:
1883:
1863:
1843:
1800:
1771:
1747:
1714:
1690:
1667:
1644:
1615:
1613:{\displaystyle n,k,d}
1583:
1553:
1533:
1492:
1450:
1430:
1395:
1375:
1315:
1313:{\displaystyle n-k+1}
1283:
1251:
1213:
1172:
1117:
1115:{\displaystyle q^{k}}
1090:
1066:
1051:and minimum distance
1046:
1026:
1002:
975:
889:
869:
830:
768:
744:
724:
694:
668:
648:
628:
605:
585:
583:{\displaystyle q^{n}}
558:
543:-ary words of length
538:
511:
438:
418:
398:
378:
336:
316:
292:
257:
176:
156:
116:
101:and minimum distance
96:
76:
56:
3218:Niederreiter, Harald
3155:, Chapman & Hall
2925:Ling & Xing 2004
2903:, p. 94 Remark 5.4.7
2901:Ling & Xing 2004
2853:Ling & Xing 2004
2752:
2702:
2682:
2660:
2639:
2610:
2572:
2497:
2468:
2448:
2406:
2090:
2070:
2050:
2023:
1994:
1962:
1942:
1899:
1872:
1852:
1812:
1782:
1760:
1730:
1721:linearly independent
1703:
1679:
1656:
1624:
1592:
1572:
1542:
1522:
1459:
1439:
1404:
1384:
1364:
1292:
1266:
1222:
1181:
1126:
1099:
1079:
1055:
1035:
1015:
991:
898:
878:
839:
780:
757:
733:
707:
699:are distinct. If we
677:
657:
637:
617:
594:
567:
547:
527:
450:
427:
407:
387:
345:
325:
305:
266:
185:
165:
145:
129:and even earlier by
105:
85:
65:
45:
2975:1988InMat..92..441B
2634: —
2400:projective geometry
1936: —
1754:parity check matrix
1745:{\displaystyle n-k}
1566: —
1380:codewords (the all-
1281:{\displaystyle n-k}
1260:parity check matrix
722:{\displaystyle d-1}
2983:10.1007/bf01393742
2767:
2738:
2688:
2666:
2645:
2632:
2616:
2596:
2554:
2483:
2454:
2430:
2377:
2076:
2056:
2036:
2009:
1980:
1948:
1934:
1905:
1878:
1858:
1838:
1795:
1766:
1742:
1709:
1685:
1662:
1639:
1610:
1578:
1564:
1548:
1528:
1513:Reed-Solomon codes
1487:
1445:
1425:
1390:
1370:
1310:
1278:
1246:
1208:
1167:
1112:
1085:
1061:
1041:
1021:
1011:with block length
997:
970:
884:
864:
825:
763:
739:
719:
689:
663:
643:
623:
600:
580:
553:
533:
506:
433:
413:
393:
373:
331:
311:
287:
252:
233:
171:
151:
111:
91:
71:
61:with block length
51:
3056:MacWilliams, F.J.
2891:, Proposition 9.2
2691:{\displaystyle G}
2669:{\displaystyle m}
2648:{\displaystyle K}
2630:
2619:{\displaystyle G}
2457:{\displaystyle N}
2344:
2253:
2188:
2120:
2079:{\displaystyle w}
2059:{\displaystyle C}
1951:{\displaystyle C}
1932:
1908:{\displaystyle d}
1881:{\displaystyle A}
1861:{\displaystyle C}
1769:{\displaystyle C}
1712:{\displaystyle C}
1688:{\displaystyle k}
1665:{\displaystyle C}
1581:{\displaystyle C}
1562:
1551:{\displaystyle k}
1531:{\displaystyle n}
1448:{\displaystyle n}
1393:{\displaystyle x}
1373:{\displaystyle q}
1088:{\displaystyle q}
1064:{\displaystyle d}
1044:{\displaystyle k}
1024:{\displaystyle n}
1000:{\displaystyle C}
887:{\displaystyle C}
766:{\displaystyle d}
742:{\displaystyle C}
666:{\displaystyle d}
646:{\displaystyle q}
626:{\displaystyle C}
603:{\displaystyle q}
556:{\displaystyle n}
536:{\displaystyle q}
436:{\displaystyle d}
416:{\displaystyle n}
396:{\displaystyle q}
341:. The expression
334:{\displaystyle y}
314:{\displaystyle x}
194:
174:{\displaystyle n}
154:{\displaystyle C}
114:{\displaystyle d}
94:{\displaystyle M}
74:{\displaystyle n}
54:{\displaystyle C}
16:(Redirected from
3277:
3251:
3213:
3174:
3156:
3147:
3127:
3105:
3082:
3051:
3033:
3024:
3023:
2994:
2993:
2958:
2952:
2946:
2940:
2934:
2928:
2922:
2916:
2910:
2904:
2898:
2892:
2886:
2880:
2874:
2868:
2862:
2856:
2850:
2844:
2843:
2823:
2776:
2774:
2773:
2768:
2766:
2765:
2760:
2747:
2745:
2744:
2741:{\displaystyle }
2739:
2697:
2695:
2694:
2689:
2675:
2673:
2672:
2667:
2654:
2652:
2651:
2646:
2635:
2625:
2623:
2622:
2617:
2605:
2603:
2602:
2597:
2563:
2561:
2560:
2555:
2550:
2549:
2531:
2530:
2518:
2517:
2492:
2490:
2489:
2484:
2482:
2481:
2476:
2463:
2461:
2460:
2455:
2442:projective space
2439:
2437:
2436:
2431:
2386:
2384:
2383:
2378:
2373:
2372:
2351:
2350:
2349:
2340:
2328:
2321:
2320:
2301:
2290:
2260:
2259:
2258:
2245:
2226:
2225:
2195:
2194:
2193:
2180:
2173:
2172:
2153:
2142:
2127:
2126:
2125:
2112:
2102:
2101:
2085:
2083:
2082:
2077:
2065:
2063:
2062:
2057:
2045:
2043:
2042:
2037:
2035:
2034:
2018:
2016:
2015:
2010:
2008:
2007:
2002:
1990:] MDS code over
1989:
1987:
1986:
1981:
1957:
1955:
1954:
1949:
1937:
1914:
1912:
1911:
1906:
1887:
1885:
1884:
1879:
1867:
1865:
1864:
1859:
1847:
1845:
1844:
1839:
1831:
1804:
1802:
1801:
1796:
1794:
1793:
1775:
1773:
1772:
1767:
1751:
1749:
1748:
1743:
1718:
1716:
1715:
1710:
1697:generator matrix
1694:
1692:
1691:
1686:
1671:
1669:
1668:
1663:
1648:
1646:
1645:
1640:
1638:
1637:
1632:
1619:
1617:
1616:
1611:
1587:
1585:
1584:
1579:
1567:
1557:
1555:
1554:
1549:
1537:
1535:
1534:
1529:
1496:
1494:
1493:
1488:
1486:
1485:
1476:
1475:
1470:
1454:
1452:
1451:
1446:
1434:
1432:
1431:
1426:
1424:
1423:
1418:
1399:
1397:
1396:
1391:
1379:
1377:
1376:
1371:
1330:Singleton (1964)
1319:
1317:
1316:
1311:
1287:
1285:
1284:
1279:
1255:
1253:
1252:
1247:
1217:
1215:
1214:
1209:
1176:
1174:
1173:
1168:
1163:
1162:
1138:
1137:
1121:
1119:
1118:
1113:
1111:
1110:
1094:
1092:
1091:
1086:
1070:
1068:
1067:
1062:
1050:
1048:
1047:
1042:
1030:
1028:
1027:
1022:
1006:
1004:
1003:
998:
979:
977:
976:
971:
966:
965:
926:
925:
913:
905:
893:
891:
890:
885:
873:
871:
870:
865:
863:
862:
834:
832:
831:
826:
772:
770:
769:
764:
751:Hamming distance
748:
746:
745:
740:
728:
726:
725:
720:
698:
696:
695:
690:
672:
670:
669:
664:
652:
650:
649:
644:
633:be an arbitrary
632:
630:
629:
624:
609:
607:
606:
601:
589:
587:
586:
581:
579:
578:
562:
560:
559:
554:
542:
540:
539:
534:
515:
513:
512:
507:
502:
501:
462:
461:
442:
440:
439:
434:
422:
420:
419:
414:
402:
400:
399:
394:
382:
380:
379:
374:
357:
356:
340:
338:
337:
332:
320:
318:
317:
312:
299:Hamming distance
296:
294:
293:
288:
261:
259:
258:
253:
232:
180:
178:
177:
172:
160:
158:
157:
152:
120:
118:
117:
112:
100:
98:
97:
92:
80:
78:
77:
72:
60:
58:
57:
52:
21:
3285:
3284:
3280:
3279:
3278:
3276:
3275:
3274:
3255:
3254:
3240:
3210:
3181:
3179:Further reading
3172:
3125:
3103:
3080:
3049:
3002:
2997:
2959:
2955:
2947:
2943:
2935:
2931:
2923:
2919:
2911:
2907:
2899:
2895:
2887:
2883:
2875:
2871:
2863:
2859:
2851:
2847:
2840:
2824:
2820:
2816:
2784:
2779:
2761:
2756:
2755:
2753:
2750:
2749:
2703:
2700:
2699:
2683:
2680:
2679:
2678:if and only if
2661:
2658:
2657:
2655:is a (spatial)
2640:
2637:
2636:
2633:
2611:
2608:
2607:
2573:
2570:
2569:
2545:
2541:
2526:
2522:
2513:
2509:
2498:
2495:
2494:
2477:
2472:
2471:
2469:
2466:
2465:
2449:
2446:
2445:
2407:
2404:
2403:
2393:
2388:
2356:
2352:
2345:
2330:
2324:
2323:
2322:
2316:
2312:
2291:
2280:
2254:
2241:
2240:
2239:
2203:
2199:
2189:
2176:
2175:
2174:
2168:
2164:
2143:
2132:
2121:
2108:
2107:
2106:
2097:
2093:
2091:
2088:
2087:
2071:
2068:
2067:
2051:
2048:
2047:
2030:
2026:
2024:
2021:
2020:
2003:
1998:
1997:
1995:
1992:
1991:
1963:
1960:
1959:
1943:
1940:
1939:
1935:
1922:
1900:
1897:
1896:
1873:
1870:
1869:
1853:
1850:
1849:
1827:
1813:
1810:
1809:
1805:is an MDS code.
1789:
1785:
1783:
1780:
1779:
1761:
1758:
1757:
1731:
1728:
1727:
1704:
1701:
1700:
1680:
1677:
1676:
1672:is an MDS code.
1657:
1654:
1653:
1633:
1628:
1627:
1625:
1622:
1621:
1593:
1590:
1589:
1573:
1570:
1569:
1565:
1543:
1540:
1539:
1523:
1520:
1519:
1481:
1477:
1471:
1466:
1465:
1460:
1457:
1456:
1440:
1437:
1436:
1419:
1414:
1413:
1405:
1402:
1401:
1385:
1382:
1381:
1365:
1362:
1361:
1354:
1346:Komamiya (1953)
1338:Komamiya (1953)
1326:
1293:
1290:
1289:
1267:
1264:
1263:
1223:
1220:
1219:
1182:
1179:
1178:
1146:
1142:
1133:
1129:
1127:
1124:
1123:
1106:
1102:
1100:
1097:
1096:
1080:
1077:
1076:
1056:
1053:
1052:
1036:
1033:
1032:
1016:
1013:
1012:
992:
989:
988:
985:
949:
945:
921:
917:
909:
901:
899:
896:
895:
879:
876:
875:
874:of them. Since
846:
842:
840:
837:
836:
781:
778:
777:
758:
755:
754:
734:
731:
730:
708:
705:
704:
678:
675:
674:
658:
655:
654:
638:
635:
634:
618:
615:
614:
595:
592:
591:
574:
570:
568:
565:
564:
548:
545:
544:
528:
525:
524:
521:
485:
481:
457:
453:
451:
448:
447:
428:
425:
424:
408:
405:
404:
388:
385:
384:
352:
348:
346:
343:
342:
326:
323:
322:
306:
303:
302:
267:
264:
263:
198:
186:
183:
182:
181:is defined as
166:
163:
162:
146:
143:
142:
139:
131:Komamiya (1953)
106:
103:
102:
86:
83:
82:
66:
63:
62:
46:
43:
42:
36:Singleton bound
28:
23:
22:
15:
12:
11:
5:
3283:
3273:
3272:
3267:
3253:
3252:
3238:
3214:
3208:
3180:
3177:
3176:
3175:
3170:
3157:
3148:
3139:(2): 116–118,
3128:
3123:
3106:
3101:
3083:
3078:
3060:Sloane, N.J.A.
3052:
3047:
3034:
3025:
3014:(3): 289–295,
3001:
2998:
2996:
2995:
2969:(3): 441–459,
2953:
2941:
2929:
2917:
2905:
2893:
2881:
2869:
2857:
2845:
2838:
2817:
2815:
2812:
2811:
2810:
2805:
2800:
2795:
2793:Griesmer bound
2790:
2783:
2780:
2764:
2759:
2748:MDS code over
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2687:
2665:
2644:
2628:
2615:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2553:
2548:
2544:
2540:
2537:
2534:
2529:
2525:
2521:
2516:
2512:
2508:
2505:
2502:
2480:
2475:
2453:
2440:be the finite
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2392:
2389:
2376:
2371:
2368:
2365:
2362:
2359:
2355:
2348:
2343:
2339:
2336:
2333:
2327:
2319:
2315:
2311:
2308:
2305:
2300:
2297:
2294:
2289:
2286:
2283:
2279:
2275:
2272:
2269:
2266:
2263:
2257:
2252:
2249:
2244:
2238:
2235:
2232:
2229:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2202:
2198:
2192:
2187:
2184:
2179:
2171:
2167:
2163:
2160:
2157:
2152:
2149:
2146:
2141:
2138:
2135:
2131:
2124:
2119:
2116:
2111:
2105:
2100:
2096:
2075:
2055:
2033:
2029:
2006:
2001:
1979:
1976:
1973:
1970:
1967:
1947:
1930:
1921:
1920:
1904:
1893:
1877:
1857:
1837:
1834:
1830:
1826:
1823:
1820:
1817:
1806:
1792:
1788:
1777:
1765:
1741:
1738:
1735:
1724:
1708:
1684:
1673:
1661:
1636:
1631:
1609:
1606:
1603:
1600:
1597:
1577:
1560:
1547:
1527:
1484:
1480:
1474:
1469:
1464:
1444:
1422:
1417:
1412:
1409:
1389:
1369:
1353:
1350:
1325:
1322:
1309:
1306:
1303:
1300:
1297:
1277:
1274:
1271:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1166:
1161:
1158:
1155:
1152:
1149:
1145:
1141:
1136:
1132:
1109:
1105:
1084:
1060:
1040:
1020:
996:
984:
981:
969:
964:
961:
958:
955:
952:
948:
944:
941:
938:
935:
932:
929:
924:
920:
916:
912:
908:
904:
883:
861:
858:
855:
852:
849:
845:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
791:
788:
785:
762:
738:
718:
715:
712:
688:
685:
682:
662:
642:
622:
599:
577:
573:
552:
532:
520:
517:
505:
500:
497:
494:
491:
488:
484:
480:
477:
474:
471:
468:
465:
460:
456:
432:
412:
392:
372:
369:
366:
363:
360:
355:
351:
330:
310:
286:
283:
280:
277:
274:
271:
251:
248:
245:
242:
239:
236:
231:
228:
225:
222:
219:
216:
213:
210:
207:
204:
201:
197:
193:
190:
170:
150:
138:
135:
110:
90:
70:
50:
26:
9:
6:
4:
3:
2:
3282:
3271:
3268:
3266:
3265:Coding theory
3263:
3262:
3260:
3249:
3245:
3241:
3239:0-521-66543-4
3235:
3231:
3227:
3223:
3219:
3215:
3211:
3209:3-540-54894-7
3205:
3201:
3197:
3193:
3192:
3187:
3186:J.H. van Lint
3183:
3182:
3173:
3171:0-19-853287-3
3167:
3163:
3158:
3154:
3149:
3146:
3142:
3138:
3134:
3129:
3126:
3124:0-387-97812-7
3120:
3116:
3112:
3107:
3104:
3102:0-471-19047-0
3098:
3094:
3093:
3088:
3084:
3081:
3079:0-444-85193-3
3075:
3071:
3067:
3066:
3061:
3057:
3053:
3050:
3048:0-521-52923-9
3044:
3040:
3035:
3031:
3026:
3022:
3017:
3013:
3009:
3004:
3003:
2992:
2988:
2984:
2980:
2976:
2972:
2968:
2964:
2963:Invent. Math.
2957:
2950:
2945:
2938:
2933:
2926:
2921:
2914:
2909:
2902:
2897:
2890:
2885:
2878:
2873:
2866:
2861:
2854:
2849:
2841:
2839:0-444-88899-3
2835:
2831:
2830:
2822:
2818:
2809:
2808:Plotkin bound
2806:
2804:
2803:Johnson bound
2801:
2799:
2798:Hamming bound
2796:
2794:
2791:
2789:
2786:
2785:
2778:
2762:
2732:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2685:
2677:
2663:
2642:
2627:
2613:
2593:
2590:
2584:
2581:
2578:
2567:
2546:
2542:
2538:
2535:
2532:
2527:
2523:
2519:
2514:
2510:
2503:
2500:
2478:
2451:
2443:
2424:
2421:
2418:
2412:
2409:
2401:
2398:
2387:
2374:
2369:
2366:
2363:
2360:
2357:
2353:
2341:
2337:
2334:
2331:
2317:
2309:
2306:
2298:
2295:
2292:
2287:
2284:
2281:
2277:
2270:
2267:
2264:
2250:
2247:
2236:
2230:
2227:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2200:
2185:
2182:
2169:
2161:
2158:
2150:
2147:
2144:
2139:
2136:
2133:
2129:
2117:
2114:
2103:
2098:
2094:
2073:
2053:
2031:
2027:
2004:
1977:
1974:
1971:
1968:
1965:
1958:be a linear [
1945:
1929:
1927:
1918:
1902:
1894:
1891:
1875:
1855:
1832:
1824:
1818:
1815:
1807:
1790:
1786:
1778:
1763:
1755:
1752:columns of a
1739:
1736:
1733:
1725:
1722:
1706:
1698:
1695:columns of a
1682:
1674:
1659:
1652:
1651:
1650:
1634:
1607:
1604:
1601:
1598:
1595:
1588:be a linear [
1575:
1559:
1545:
1525:
1516:
1514:
1509:
1506:
1504:
1500:
1482:
1472:
1442:
1420:
1410:
1407:
1387:
1367:
1359:
1349:
1347:
1343:
1339:
1335:
1331:
1321:
1307:
1304:
1301:
1298:
1295:
1275:
1272:
1269:
1261:
1256:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1164:
1159:
1156:
1153:
1150:
1147:
1143:
1139:
1134:
1130:
1107:
1103:
1082:
1074:
1058:
1038:
1018:
1010:
994:
980:
967:
962:
959:
956:
953:
950:
946:
942:
936:
933:
930:
922:
918:
914:
906:
881:
859:
856:
853:
850:
847:
843:
822:
819:
816:
813:
810:
807:
804:
798:
795:
792:
786:
783:
774:
760:
752:
736:
716:
713:
710:
702:
686:
683:
680:
660:
640:
620:
611:
597:
575:
571:
550:
530:
516:
503:
498:
495:
492:
489:
486:
482:
478:
472:
469:
466:
458:
454:
444:
430:
410:
390:
367:
364:
361:
353:
349:
328:
308:
300:
281:
278:
275:
269:
246:
243:
240:
234:
226:
223:
220:
217:
214:
211:
208:
205:
202:
191:
188:
168:
148:
134:
132:
128:
124:
108:
88:
68:
48:
41:
37:
33:
32:coding theory
19:
3270:Inequalities
3221:
3190:
3161:
3152:
3136:
3132:
3110:
3091:
3064:
3038:
3029:
3011:
3007:
2966:
2962:
2956:
2944:
2932:
2920:
2908:
2896:
2889:Vermani 1996
2884:
2872:
2860:
2848:
2828:
2821:
2629:
2394:
1931:
1923:
1620:] code over
1561:
1517:
1510:
1507:
1502:
1357:
1355:
1334:Joshi (1958)
1327:
1257:
1073:finite field
1031:, dimension
986:
983:Linear codes
775:
612:
522:
445:
140:
127:Joshi (1958)
125:. proved by
122:
35:
29:
3087:Pless, Vera
2568:. Form the
1890:nonsingular
1505:MDS codes.
1342:Welsh (1988
1009:linear code
3259:Categories
3248:0971.11033
3000:References
2949:Roman 1992
2937:Roman 1992
2877:Pless 1998
2865:Roman 1992
2066:of weight
1895:Given any
1499:dual codes
123:Joshibound
40:block code
3226:Cambridge
2991:120077696
2730:−
2591:×
2536:…
2367:−
2361:−
2335:−
2307:−
2296:−
2278:∑
2268:−
2228:−
2220:−
2208:−
2159:−
2148:−
2130:∑
1791:⊥
1737:−
1411:∈
1400:word for
1352:MDS codes
1299:−
1273:−
1235:−
1229:≤
1194:−
1188:≤
1151:−
1140:≤
1071:over the
954:−
943:≤
915:≤
851:−
811:−
796:−
787:−
753:at least
714:−
684:∈
490:−
479:≤
224:≠
212:∈
3188:(1992).
3089:(1998),
3062:(1977),
2951:, p. 240
2915:, Ch. 11
2867:, p. 175
2782:See also
1177:so that
701:puncture
613:Now let
301:between
2971:Bibcode
2927:, p. 94
2879:, p. 26
2855:, p. 93
2631:Theorem
2606:matrix
2086:, then
1933:Theorem
1917:support
1563:Theorem
1503:trivial
1324:History
297:is the
81:, size
3246:
3236:
3206:
3168:
3121:
3099:
3076:
3070:33, 37
3045:
2989:
2836:
2493:. Let
2402:. Let
2397:finite
262:where
34:, the
3032:: 437
2987:S2CID
2814:Notes
2019:. If
1075:with
1007:is a
749:have
519:Proof
3234:ISBN
3204:ISBN
3166:ISBN
3119:ISBN
3097:ISBN
3074:ISBN
3043:ISBN
2834:ISBN
2676:-arc
1938:Let
1756:for
1726:Any
1719:are
1699:for
1675:Any
1568:Let
1538:and
321:and
3244:Zbl
3196:GTM
3141:doi
3115:GTM
3016:doi
2979:doi
1888:is
1808:If
1262:is
987:If
563:is
196:min
30:In
3261::
3242:.
3232:.
3228::
3202:.
3200:61
3194:.
3137:10
3135:,
3113:,
3072:,
3058:;
3010:,
2985:,
2977:,
2967:92
2965:,
2777:.
1348:.
1320:.
1244:1.
443:.
133:.
3250:.
3212:.
3143::
3018::
3012:1
2981::
2973::
2842:.
2763:q
2758:F
2736:]
2733:N
2727:m
2724:,
2721:1
2718:+
2715:N
2712:,
2709:m
2706:[
2686:G
2664:m
2643:K
2614:G
2594:m
2588:)
2585:1
2582:+
2579:N
2576:(
2552:}
2547:m
2543:P
2539:,
2533:,
2528:2
2524:P
2520:,
2515:1
2511:P
2507:{
2504:=
2501:K
2479:q
2474:F
2452:N
2428:)
2425:q
2422:,
2419:N
2416:(
2413:G
2410:P
2375:.
2370:j
2364:d
2358:w
2354:q
2347:)
2342:j
2338:1
2332:w
2326:(
2318:j
2314:)
2310:1
2304:(
2299:d
2293:w
2288:0
2285:=
2282:j
2274:)
2271:1
2265:q
2262:(
2256:)
2251:w
2248:n
2243:(
2237:=
2234:)
2231:1
2223:j
2217:1
2214:+
2211:d
2205:w
2201:q
2197:(
2191:)
2186:j
2183:w
2178:(
2170:j
2166:)
2162:1
2156:(
2151:d
2145:w
2140:0
2137:=
2134:j
2123:)
2118:w
2115:n
2110:(
2104:=
2099:w
2095:A
2074:w
2054:C
2032:w
2028:A
2005:q
2000:F
1978:d
1975:,
1972:k
1969:,
1966:n
1946:C
1903:d
1892:.
1876:A
1856:C
1836:)
1833:A
1829:|
1825:I
1822:(
1819:=
1816:G
1787:C
1764:C
1740:k
1734:n
1723:.
1707:C
1683:k
1660:C
1635:q
1630:F
1608:d
1605:,
1602:k
1599:,
1596:n
1576:C
1546:k
1526:n
1483:n
1479:)
1473:q
1468:F
1463:(
1443:n
1421:q
1416:F
1408:x
1388:x
1368:q
1308:1
1305:+
1302:k
1296:n
1276:k
1270:n
1241:+
1238:k
1232:n
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1206:,
1203:1
1200:+
1197:d
1191:n
1185:k
1165:,
1160:1
1157:+
1154:d
1148:n
1144:q
1135:k
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1108:k
1104:q
1083:q
1059:d
1039:k
1019:n
995:C
968:.
963:1
960:+
957:d
951:n
947:q
940:)
937:d
934:,
931:n
928:(
923:q
919:A
911:|
907:C
903:|
882:C
860:1
857:+
854:d
848:n
844:q
823:,
820:1
817:+
814:d
808:n
805:=
802:)
799:1
793:d
790:(
784:n
761:d
737:C
717:1
711:d
687:C
681:c
661:d
641:q
621:C
598:q
576:n
572:q
551:n
531:q
504:.
499:1
496:+
493:d
487:n
483:q
476:)
473:d
470:,
467:n
464:(
459:q
455:A
431:d
411:n
391:q
371:)
368:d
365:,
362:n
359:(
354:q
350:A
329:y
309:x
285:)
282:y
279:,
276:x
273:(
270:d
250:)
247:y
244:,
241:x
238:(
235:d
230:}
227:y
221:x
218::
215:C
209:y
206:,
203:x
200:{
192:=
189:d
169:n
149:C
109:d
89:M
69:n
49:C
20:)
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