1733:
1872:
830:
1379:
2115:
977:
888:
1570:
2212:
1562:
1744:
2020:
1972:
1503:
1277:
421:
318:
503:
452:
282:
211:
576:
1932:
1423:
695:
1037:
1221:
345:
619:
1463:
1443:
1241:
1189:
1169:
1141:
1121:
1101:
1081:
1061:
1008:
916:
662:
639:
596:
523:
472:
385:
365:
251:
231:
706:
1285:
2442:
990:
will decode it correctly (i.e., it decodes the received word as the codeword that was sent). Thus the code is said to be capable of correcting
2646:
2060:
2592:
924:
2642:
841:
1123:
be the number of words in each ball (in other words, the volume of the ball). A word that is in such a ball can deviate in at most
1728:{\displaystyle A_{q}(n,d)\times m=A_{q}(n,d)\times {\begin{matrix}\sum _{k=0}^{t}{\binom {n}{k}}(q-1)^{k}\end{matrix}}\leq q^{n}.}
2226:
perfect codes. In 1973, Tietäväinen proved that any non-trivial perfect code over a prime-power alphabet has the parameters of a
1469:
of the words in these balls centered at codewords, results in a set of words, each counted precisely once, that is a subset of
2548:
2179:
2529:
2475:
2452:
2428:
2398:
1508:
122:, and finally decoded by the receiver. The decoding process interprets a garbled codeword, referred to as simply a
1867:{\displaystyle A_{q}(n,d)\leq {\frac {q^{n}}{\begin{matrix}\sum _{k=0}^{t}{\binom {n}{k}}(q-1)^{k}\end{matrix}}}.}
2779:
2711:
2585:
2521:
2467:
1989:
1941:
1472:
1246:
390:
287:
2706:
17:
2231:
477:
426:
256:
185:
2696:
2650:
2638:
2352:
P. J. Cameron; J. A. Thas; S. E. Payne (1976). "Polarities of generalized hexagons and perfect codes".
2218:, where each symbol of the message is repeated an odd fixed number of times to obtain a codeword where
2686:
2578:
987:
983:
99:
79:
2701:
536:
2390:
2270:
1895:
1386:
667:
2800:
2660:
2290:
2691:
1148:
119:
75:
1016:
2257:+1 cover the space, possibly with some overlaps. Another way to say this is that a code is
1194:
323:
8:
2753:
2615:
825:{\displaystyle \ A_{q}(n,d)\leq {\frac {q^{n}}{\sum _{k=0}^{t}{\binom {n}{k}}(q-1)^{k}}}}
601:
67:
2514:
2417:
2408:
2383:
2369:
1466:
1448:
1428:
1226:
1174:
1154:
1144:
1126:
1106:
1086:
1066:
1046:
1040:
993:
901:
647:
624:
581:
508:
457:
370:
350:
236:
216:
2176:. Examples include codes that have only one codeword, and codes that are the whole of
1374:{\displaystyle m={\begin{matrix}\sum _{k=0}^{t}{\binom {n}{k}}(q-1)^{k}\end{matrix}}.}
2758:
2655:
2525:
2471:
2448:
2424:
2394:
2373:
1465:, the greatest number of balls with no two balls having a word in common. Taking the
142:
2601:
2557:
2497:
2361:
642:
35:
2485:
2295:
165:
words can be received because the noisy channel might distort one or more of the
2737:
2716:
2678:
2665:
2630:
2275:
94:
An original message and an encoded version are both composed in an alphabet of
63:
59:
2237:
A perfect code may be interpreted as one in which the balls of
Hamming radius
2794:
2774:
2562:
2543:
2539:
2509:
2285:
2280:
2051:
39:
2227:
1083:. Every pair of these balls (Hamming spheres) are non-intersecting by the
2620:
2412:
31:
2438:
2365:
1885:
71:
47:
2501:
2110:{\displaystyle t\,=\,\left\lfloor {\frac {1}{2}}(d-1)\right\rfloor }
1147:, which is a codeword. The number of such words is then obtained by
505:
makes no difference to the result, provided the alphabet is of size
118:-letter codeword by an encoding algorithm, transmitted over a noisy
82:
are embedded. A code that attains the
Hamming bound is said to be a
2570:
2351:
2164:. The case of equality means that the Hamming bound is attained.
972:{\displaystyle t=\left\lfloor {\frac {1}{2}}(d-1)\right\rfloor }
70:
of all possible words. It gives an important limitation on the
2261:
if its covering radius is one greater than its packing radius.
883:{\displaystyle t=\left\lfloor {\frac {d-1}{2}}\right\rfloor .}
2057:
From the proof of the
Hamming bound, it can be seen that for
664:
between elements of the block code (necessarily positive for
2253:
centered on codewords are disjoint and the balls of radius
2486:"On the nonexistence of perfect codes over finite fields"
2407:
89:
2308:
2183:
1788:
1637:
1296:
2182:
2063:
1992:
1944:
1898:
1877:
1747:
1573:
1511:
1475:
1451:
1431:
1389:
1288:
1249:
1229:
1197:
1177:
1157:
1129:
1109:
1089:
1069:
1049:
1019:
996:
927:
904:
844:
709:
670:
650:
627:
604:
584:
539:
511:
480:
460:
429:
393:
373:
353:
326:
290:
259:
239:
219:
188:
2444:
Introduction to the Theory of Error-Correcting Codes
2207:{\displaystyle \scriptstyle {\mathcal {A}}_{q}^{n}}
2513:
2416:
2382:
2241:centered on codewords exactly fill out the space (
2206:
2109:
2014:
1966:
1926:
1866:
1727:
1556:
1497:
1457:
1437:
1417:
1373:
1271:
1235:
1215:
1183:
1163:
1135:
1115:
1095:
1075:
1055:
1031:
1002:
971:
910:
882:
824:
689:
656:
633:
613:
590:
570:
517:
497:
466:
446:
415:
379:
359:
339:
312:
276:
245:
225:
205:
2792:
2524:. Vol. 86 (2nd ed.). Springer-Verlag.
2222:= 2. All of these examples are often called the
2172:Codes that attain the Hamming bound are called
1557:{\displaystyle |{\mathcal {A}}_{q}^{n}|=q^{n}}
1425:is the (maximum) total number of codewords in
1191:components of a codeword to deviate to one of
2586:
2245:is the covering radius = packing radius). A
1828:
1815:
1677:
1664:
1336:
1323:
791:
778:
161:valid codewords are possible, but any one of
46:is a limit on the parameters of an arbitrary
2470:, vol. 134, New York: Springer-Verlag,
2249:is one in which the balls of Hamming radius
2022:is contained in at least one ball of radius
347:distinct strings in this set of strings.) A
2483:
2314:
1223:possible other values (recall, the code is
2593:
2579:
177:
114:letters. The message is converted into an
2561:
2071:
2067:
982:errors are made during transmission of a
172:
106:letters. The original message (of length
2538:
2508:
169:letters when a codeword is transmitted.
141:, and each message can be regarded as a
233:elements. The set of strings of length
27:Limit on the parameters of a block code
14:
2793:
2015:{\displaystyle {\mathcal {A}}_{q}^{n}}
1967:{\displaystyle {\mathcal {A}}_{q}^{n}}
1498:{\displaystyle {\mathcal {A}}_{q}^{n}}
1272:{\displaystyle {\mathcal {A}}_{q}^{n}}
578:denote the maximum possible size of a
474:elements. (The choice of alphabet set
416:{\displaystyle {\mathcal {A}}_{q}^{n}}
313:{\displaystyle {\mathcal {A}}_{q}^{n}}
126:, as the valid codeword "nearest" the
2574:
2549:Rocky Mountain Journal of Mathematics
2461:
2437:
2335:
2042:such that the set of balls of radius
528:
2600:
2419:The Theory of Error-Correcting Codes
2380:
1143:components from those of the ball's
90:Background on error-correcting codes
78:can utilize the space in which its
58:from an interpretation in terms of
24:
2214:. Another example is given by the
2187:
1996:
1948:
1878:Covering radius and packing radius
1819:
1668:
1520:
1479:
1327:
1253:
898:It follows from the definition of
782:
498:{\displaystyle {\mathcal {A}}_{q}}
484:
447:{\displaystyle {\mathcal {A}}_{q}}
433:
397:
294:
277:{\displaystyle {\mathcal {A}}_{q}}
263:
206:{\displaystyle {\mathcal {A}}_{q}}
192:
149:. The encoding scheme converts an
133:Mathematically, there are exactly
25:
2812:
2167:
1103:-error-correcting property. Let
2385:A First Course In Coding Theory
1445:, and so, by the definition of
2722:Sphere-packing (Hamming) bound
2329:
2320:
2099:
2087:
1921:
1909:
1847:
1834:
1770:
1758:
1696:
1683:
1630:
1618:
1596:
1584:
1537:
1513:
1412:
1400:
1355:
1342:
1210:
1198:
961:
949:
810:
797:
735:
723:
565:
553:
387:is a subset of the strings of
13:
1:
2516:Introduction to Coding Theory
2464:Coding and Information Theory
2345:
2046:centered at each codeword of
2026:centered at each codeword of
157:-dimensional vector. Exactly
2326:McWilliams and Sloane, p. 19
700:Then, the Hamming bound is:
571:{\displaystyle \ A_{q}(n,d)}
153:-dimensional vector into an
137:possible messages of length
7:
2544:"A survey of perfect codes"
2264:
2152:and if equality holds then
1986:such that every element of
454:is any alphabet set having
10:
2817:
1927:{\displaystyle A_{q}(n,d)}
1883:
1418:{\displaystyle A_{q}(n,d)}
690:{\displaystyle q^{n}>1}
367:-ary block code of length
50:: it is also known as the
2767:
2746:
2730:
2677:
2629:
2608:
2447:. John Wiley & Sons.
1982:is the smallest value of
1243:-ary: it takes values in
988:minimum distance decoding
423:, where the alphabet set
213:is a set of symbols with
130:-letter received string.
2647:isosceles right triangle
2563:10.1216/RMJ-1975-5-2-199
2484:Tietäväinen, A. (1973).
2301:
2038:is the largest value of
893:
2391:Oxford University Press
2271:Gilbert-Varshamov bound
178:Preliminary definitions
2661:Circle packing theorem
2291:Rate-distortion theory
2208:
2111:
2016:
1968:
1928:
1868:
1811:
1729:
1660:
1558:
1499:
1459:
1439:
1419:
1375:
1319:
1273:
1237:
1217:
1185:
1165:
1137:
1117:
1097:
1077:
1057:
1033:
1032:{\displaystyle c\in C}
1004:
973:
912:
884:
826:
774:
691:
658:
635:
615:
592:
572:
519:
499:
468:
448:
417:
381:
361:
341:
314:
278:
247:
227:
207:
173:Statement of the bound
2209:
2112:
2017:
1969:
1929:
1869:
1791:
1730:
1640:
1559:
1500:
1460:
1440:
1420:
1376:
1299:
1274:
1238:
1218:
1216:{\displaystyle (q-1)}
1186:
1166:
1138:
1118:
1098:
1078:
1058:
1034:
1005:
974:
913:
885:
827:
754:
692:
659:
636:
616:
593:
573:
520:
500:
469:
449:
418:
382:
362:
342:
340:{\displaystyle q^{n}}
315:
279:
248:
228:
208:
76:error-correcting code
2643:equilateral triangle
2180:
2061:
1990:
1942:
1896:
1745:
1571:
1509:
1473:
1449:
1429:
1387:
1286:
1247:
1227:
1195:
1175:
1155:
1127:
1107:
1087:
1067:
1047:
1017:
994:
925:
902:
842:
707:
668:
648:
625:
602:
582:
537:
509:
478:
458:
427:
391:
371:
351:
324:
288:
257:
253:on the alphabet set
237:
217:
186:
52:sphere-packing bound
2780:Slothouber–Graatsma
2354:Geometriae Dedicata
2202:
2011:
1963:
1535:
1494:
1268:
614:{\displaystyle \ C}
412:
309:
2490:SIAM J. Appl. Math
2462:Roman, S. (1992),
2409:MacWilliams, F. J.
2366:10.1007/BF00150782
2247:quasi-perfect code
2204:
2203:
2184:
2107:
2012:
1993:
1964:
1945:
1924:
1864:
1858:
1725:
1707:
1554:
1517:
1495:
1476:
1455:
1435:
1415:
1371:
1366:
1269:
1250:
1233:
1213:
1181:
1161:
1133:
1113:
1093:
1073:
1053:
1029:
1013:For each codeword
1000:
969:
908:
880:
822:
687:
654:
631:
611:
588:
568:
529:Defining the bound
515:
495:
464:
444:
413:
394:
377:
357:
337:
310:
291:
274:
243:
223:
203:
110:) is shorter than
38:, in the field of
2788:
2787:
2747:Other 3-D packing
2731:Other 2-D packing
2656:Apollonian gasket
2423:. North-Holland.
2381:Hill, R. (1988).
2085:
1859:
1826:
1675:
1458:{\displaystyle t}
1438:{\displaystyle C}
1334:
1236:{\displaystyle q}
1184:{\displaystyle n}
1164:{\displaystyle t}
1136:{\displaystyle t}
1116:{\displaystyle m}
1096:{\displaystyle t}
1076:{\displaystyle c}
1056:{\displaystyle t}
1003:{\displaystyle t}
947:
918:that if at most
911:{\displaystyle d}
871:
820:
789:
712:
657:{\displaystyle d}
634:{\displaystyle n}
607:
591:{\displaystyle q}
542:
518:{\displaystyle q}
467:{\displaystyle q}
380:{\displaystyle n}
360:{\displaystyle q}
246:{\displaystyle n}
226:{\displaystyle q}
16:(Redirected from
2808:
2669:
2609:Abstract packing
2602:Packing problems
2595:
2588:
2581:
2572:
2571:
2567:
2565:
2535:
2519:
2505:
2480:
2458:
2434:
2422:
2404:
2388:
2377:
2339:
2333:
2327:
2324:
2318:
2315:Tietäväinen 1973
2312:
2213:
2211:
2210:
2205:
2201:
2196:
2191:
2190:
2116:
2114:
2113:
2108:
2106:
2102:
2086:
2078:
2021:
2019:
2018:
2013:
2010:
2005:
2000:
1999:
1973:
1971:
1970:
1965:
1962:
1957:
1952:
1951:
1933:
1931:
1930:
1925:
1908:
1907:
1873:
1871:
1870:
1865:
1860:
1855:
1854:
1833:
1832:
1831:
1818:
1810:
1805:
1787:
1786:
1777:
1757:
1756:
1734:
1732:
1731:
1726:
1721:
1720:
1708:
1704:
1703:
1682:
1681:
1680:
1667:
1659:
1654:
1617:
1616:
1583:
1582:
1563:
1561:
1560:
1555:
1553:
1552:
1540:
1534:
1529:
1524:
1523:
1516:
1504:
1502:
1501:
1496:
1493:
1488:
1483:
1482:
1464:
1462:
1461:
1456:
1444:
1442:
1441:
1436:
1424:
1422:
1421:
1416:
1399:
1398:
1380:
1378:
1377:
1372:
1367:
1363:
1362:
1341:
1340:
1339:
1326:
1318:
1313:
1278:
1276:
1275:
1270:
1267:
1262:
1257:
1256:
1242:
1240:
1239:
1234:
1222:
1220:
1219:
1214:
1190:
1188:
1187:
1182:
1170:
1168:
1167:
1162:
1142:
1140:
1139:
1134:
1122:
1120:
1119:
1114:
1102:
1100:
1099:
1094:
1082:
1080:
1079:
1074:
1062:
1060:
1059:
1054:
1043:of fixed radius
1038:
1036:
1035:
1030:
1009:
1007:
1006:
1001:
978:
976:
975:
970:
968:
964:
948:
940:
917:
915:
914:
909:
889:
887:
886:
881:
876:
872:
867:
856:
831:
829:
828:
823:
821:
819:
818:
817:
796:
795:
794:
781:
773:
768:
752:
751:
742:
722:
721:
710:
696:
694:
693:
688:
680:
679:
663:
661:
660:
655:
643:Hamming distance
640:
638:
637:
632:
620:
618:
617:
612:
605:
598:-ary block code
597:
595:
594:
589:
577:
575:
574:
569:
552:
551:
540:
524:
522:
521:
516:
504:
502:
501:
496:
494:
493:
488:
487:
473:
471:
470:
465:
453:
451:
450:
445:
443:
442:
437:
436:
422:
420:
419:
414:
411:
406:
401:
400:
386:
384:
383:
378:
366:
364:
363:
358:
346:
344:
343:
338:
336:
335:
319:
317:
316:
311:
308:
303:
298:
297:
283:
281:
280:
275:
273:
272:
267:
266:
252:
250:
249:
244:
232:
230:
229:
224:
212:
210:
209:
204:
202:
201:
196:
195:
182:An alphabet set
36:computer science
21:
2816:
2815:
2811:
2810:
2809:
2807:
2806:
2805:
2791:
2790:
2789:
2784:
2763:
2742:
2726:
2673:
2667:
2666:Tammes problem
2625:
2604:
2599:
2540:van Lint, J. H.
2532:
2510:van Lint, J. H.
2502:10.1137/0124010
2478:
2455:
2431:
2401:
2348:
2343:
2342:
2334:
2330:
2325:
2321:
2313:
2309:
2304:
2296:Singleton bound
2267:
2197:
2192:
2186:
2185:
2181:
2178:
2177:
2170:
2077:
2076:
2072:
2062:
2059:
2058:
2006:
2001:
1995:
1994:
1991:
1988:
1987:
1976:covering radius
1958:
1953:
1947:
1946:
1943:
1940:
1939:
1903:
1899:
1897:
1894:
1893:
1888:
1880:
1857:
1856:
1850:
1846:
1827:
1814:
1813:
1812:
1806:
1795:
1782:
1778:
1776:
1752:
1748:
1746:
1743:
1742:
1716:
1712:
1706:
1705:
1699:
1695:
1676:
1663:
1662:
1661:
1655:
1644:
1636:
1612:
1608:
1578:
1574:
1572:
1569:
1568:
1564:words) and so:
1548:
1544:
1536:
1530:
1525:
1519:
1518:
1512:
1510:
1507:
1506:
1489:
1484:
1478:
1477:
1474:
1471:
1470:
1450:
1447:
1446:
1430:
1427:
1426:
1394:
1390:
1388:
1385:
1384:
1365:
1364:
1358:
1354:
1335:
1322:
1321:
1320:
1314:
1303:
1295:
1287:
1284:
1283:
1263:
1258:
1252:
1251:
1248:
1245:
1244:
1228:
1225:
1224:
1196:
1193:
1192:
1176:
1173:
1172:
1156:
1153:
1152:
1128:
1125:
1124:
1108:
1105:
1104:
1088:
1085:
1084:
1068:
1065:
1064:
1048:
1045:
1044:
1018:
1015:
1014:
995:
992:
991:
939:
938:
934:
926:
923:
922:
903:
900:
899:
896:
857:
855:
851:
843:
840:
839:
813:
809:
790:
777:
776:
775:
769:
758:
753:
747:
743:
741:
717:
713:
708:
705:
704:
675:
671:
669:
666:
665:
649:
646:
645:
626:
623:
622:
603:
600:
599:
583:
580:
579:
547:
543:
538:
535:
534:
531:
510:
507:
506:
489:
483:
482:
481:
479:
476:
475:
459:
456:
455:
438:
432:
431:
430:
428:
425:
424:
407:
402:
396:
395:
392:
389:
388:
372:
369:
368:
352:
349:
348:
331:
327:
325:
322:
321:
304:
299:
293:
292:
289:
286:
285:
268:
262:
261:
260:
258:
255:
254:
238:
235:
234:
218:
215:
214:
197:
191:
190:
189:
187:
184:
183:
180:
175:
92:
74:with which any
28:
23:
22:
15:
12:
11:
5:
2814:
2804:
2803:
2786:
2785:
2783:
2782:
2777:
2771:
2769:
2765:
2764:
2762:
2761:
2756:
2750:
2748:
2744:
2743:
2741:
2740:
2738:Square packing
2734:
2732:
2728:
2727:
2725:
2724:
2719:
2717:Kissing number
2714:
2709:
2704:
2699:
2694:
2689:
2683:
2681:
2679:Sphere packing
2675:
2674:
2672:
2671:
2663:
2658:
2653:
2635:
2633:
2631:Circle packing
2627:
2626:
2624:
2623:
2618:
2612:
2610:
2606:
2605:
2598:
2597:
2590:
2583:
2575:
2569:
2568:
2556:(2): 199–224.
2536:
2530:
2506:
2481:
2476:
2459:
2453:
2435:
2429:
2405:
2399:
2378:
2360:(4): 525–528.
2347:
2344:
2341:
2340:
2328:
2319:
2306:
2305:
2303:
2300:
2299:
2298:
2293:
2288:
2283:
2278:
2276:Griesmer bound
2273:
2266:
2263:
2200:
2195:
2189:
2169:
2166:
2142:
2141:
2140:
2139:
2105:
2101:
2098:
2095:
2092:
2089:
2084:
2081:
2075:
2070:
2066:
2032:packing radius
2009:
2004:
1998:
1961:
1956:
1950:
1923:
1920:
1917:
1914:
1911:
1906:
1902:
1890:
1889:
1884:Main article:
1879:
1876:
1875:
1874:
1863:
1853:
1849:
1845:
1842:
1839:
1836:
1830:
1825:
1822:
1817:
1809:
1804:
1801:
1798:
1794:
1790:
1789:
1785:
1781:
1775:
1772:
1769:
1766:
1763:
1760:
1755:
1751:
1736:
1735:
1724:
1719:
1715:
1711:
1702:
1698:
1694:
1691:
1688:
1685:
1679:
1674:
1671:
1666:
1658:
1653:
1650:
1647:
1643:
1639:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1615:
1611:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1581:
1577:
1551:
1547:
1543:
1539:
1533:
1528:
1522:
1515:
1492:
1487:
1481:
1454:
1434:
1414:
1411:
1408:
1405:
1402:
1397:
1393:
1382:
1381:
1370:
1361:
1357:
1353:
1350:
1347:
1344:
1338:
1333:
1330:
1325:
1317:
1312:
1309:
1306:
1302:
1298:
1297:
1294:
1291:
1266:
1261:
1255:
1232:
1212:
1209:
1206:
1203:
1200:
1180:
1160:
1132:
1112:
1092:
1072:
1052:
1028:
1025:
1022:
999:
980:
979:
967:
963:
960:
957:
954:
951:
946:
943:
937:
933:
930:
907:
895:
892:
891:
890:
879:
875:
870:
866:
863:
860:
854:
850:
847:
833:
832:
816:
812:
808:
805:
802:
799:
793:
788:
785:
780:
772:
767:
764:
761:
757:
750:
746:
740:
737:
734:
731:
728:
725:
720:
716:
686:
683:
678:
674:
653:
630:
610:
587:
567:
564:
561:
558:
555:
550:
546:
530:
527:
514:
492:
486:
463:
441:
435:
410:
405:
399:
376:
356:
334:
330:
307:
302:
296:
271:
265:
242:
222:
200:
194:
179:
176:
174:
171:
98:letters. Each
91:
88:
64:Hamming metric
26:
9:
6:
4:
3:
2:
2813:
2802:
2801:Coding theory
2799:
2798:
2796:
2781:
2778:
2776:
2773:
2772:
2770:
2766:
2760:
2757:
2755:
2752:
2751:
2749:
2745:
2739:
2736:
2735:
2733:
2729:
2723:
2720:
2718:
2715:
2713:
2712:Close-packing
2710:
2708:
2707:In a cylinder
2705:
2703:
2700:
2698:
2695:
2693:
2690:
2688:
2685:
2684:
2682:
2680:
2676:
2670:
2664:
2662:
2659:
2657:
2654:
2652:
2648:
2644:
2640:
2637:
2636:
2634:
2632:
2628:
2622:
2619:
2617:
2614:
2613:
2611:
2607:
2603:
2596:
2591:
2589:
2584:
2582:
2577:
2576:
2573:
2564:
2559:
2555:
2551:
2550:
2545:
2541:
2537:
2533:
2531:3-540-54894-7
2527:
2523:
2518:
2517:
2511:
2507:
2503:
2499:
2495:
2491:
2487:
2482:
2479:
2477:0-387-97812-7
2473:
2469:
2465:
2460:
2456:
2454:0-471-08684-3
2450:
2446:
2445:
2440:
2436:
2432:
2430:0-444-85193-3
2426:
2421:
2420:
2414:
2413:N.J.A. Sloane
2410:
2406:
2402:
2400:0-19-853803-0
2396:
2392:
2387:
2386:
2379:
2375:
2371:
2367:
2363:
2359:
2355:
2350:
2349:
2338:, p. 140
2337:
2332:
2323:
2316:
2311:
2307:
2297:
2294:
2292:
2289:
2287:
2286:Plotkin bound
2284:
2282:
2281:Johnson bound
2279:
2277:
2274:
2272:
2269:
2268:
2262:
2260:
2259:quasi-perfect
2256:
2252:
2248:
2244:
2240:
2235:
2233:
2229:
2225:
2221:
2217:
2198:
2193:
2175:
2174:perfect codes
2168:Perfect codes
2165:
2163:
2159:
2155:
2151:
2147:
2137:
2133:
2129:
2125:
2122:
2121:
2120:
2119:
2118:
2103:
2096:
2093:
2090:
2082:
2079:
2073:
2068:
2064:
2055:
2053:
2050:are mutually
2049:
2045:
2041:
2037:
2033:
2029:
2025:
2007:
2002:
1985:
1981:
1977:
1959:
1954:
1938:(a subset of
1937:
1918:
1915:
1912:
1904:
1900:
1887:
1882:
1881:
1861:
1851:
1843:
1840:
1837:
1823:
1820:
1807:
1802:
1799:
1796:
1792:
1783:
1779:
1773:
1767:
1764:
1761:
1753:
1749:
1741:
1740:
1739:
1722:
1717:
1713:
1709:
1700:
1692:
1689:
1686:
1672:
1669:
1656:
1651:
1648:
1645:
1641:
1633:
1627:
1624:
1621:
1613:
1609:
1605:
1602:
1599:
1593:
1590:
1587:
1579:
1575:
1567:
1566:
1565:
1549:
1545:
1541:
1531:
1526:
1490:
1485:
1468:
1452:
1432:
1409:
1406:
1403:
1395:
1391:
1368:
1359:
1351:
1348:
1345:
1331:
1328:
1315:
1310:
1307:
1304:
1300:
1292:
1289:
1282:
1281:
1280:
1264:
1259:
1230:
1207:
1204:
1201:
1178:
1158:
1150:
1146:
1130:
1110:
1090:
1070:
1050:
1042:
1039:, consider a
1026:
1023:
1020:
1011:
997:
989:
985:
965:
958:
955:
952:
944:
941:
935:
931:
928:
921:
920:
919:
905:
877:
873:
868:
864:
861:
858:
852:
848:
845:
838:
837:
836:
814:
806:
803:
800:
786:
783:
770:
765:
762:
759:
755:
748:
744:
738:
732:
729:
726:
718:
714:
703:
702:
701:
698:
684:
681:
676:
672:
651:
644:
628:
608:
585:
562:
559:
556:
548:
544:
526:
512:
490:
461:
439:
408:
403:
374:
354:
332:
328:
320:. (There are
305:
300:
269:
240:
220:
198:
170:
168:
164:
160:
156:
152:
148:
144:
140:
136:
131:
129:
125:
121:
117:
113:
109:
105:
101:
97:
87:
85:
81:
77:
73:
69:
65:
61:
60:packing balls
57:
53:
49:
45:
44:Hamming bound
41:
40:coding theory
37:
33:
19:
2721:
2649: /
2645: /
2641: /
2553:
2547:
2515:
2493:
2489:
2463:
2443:
2418:
2384:
2357:
2353:
2331:
2322:
2310:
2258:
2254:
2250:
2246:
2242:
2238:
2236:
2228:Hamming code
2223:
2219:
2216:repeat codes
2215:
2173:
2171:
2161:
2157:
2153:
2149:
2145:
2143:
2135:
2131:
2127:
2123:
2056:
2047:
2043:
2039:
2035:
2031:
2027:
2023:
1983:
1979:
1975:
1935:
1891:
1737:
1383:
1012:
981:
897:
834:
699:
641:and minimum
532:
284:are denoted
181:
166:
162:
158:
154:
150:
146:
138:
134:
132:
127:
123:
115:
111:
107:
103:
95:
93:
84:perfect code
83:
56:volume bound
55:
51:
43:
29:
18:Perfect code
2754:Tetrahedron
2697:In a sphere
2668:(on sphere)
2639:In a circle
2144:Therefore,
2117:, we have:
32:mathematics
2687:Apollonian
2346:References
2336:Roman 1992
2232:Golay code
1886:Delone set
621:of length
145:of length
80:code words
72:efficiency
48:block code
2759:Ellipsoid
2702:In a cube
2496:: 88–96.
2439:Pless, V.
2374:121071671
2094:−
1841:−
1793:∑
1774:≤
1710:≤
1690:−
1642:∑
1634:×
1600:×
1349:−
1301:∑
1279:). Thus,
1205:−
1024:∈
956:−
862:−
804:−
756:∑
739:≤
102:contains
100:code word
66:into the
2795:Category
2542:(1975).
2512:(1992).
2441:(1982).
2415:(1977).
2265:See also
2104:⌋
2074:⌊
2052:disjoint
1738:Whence:
1149:choosing
1010:errors.
984:codeword
966:⌋
936:⌊
874:⌋
853:⌊
2768:Puzzles
2224:trivial
1974:), the
1892:For an
1505:(where
1171:of the
1063:around
120:channel
62:in the
54:or the
2775:Conway
2692:Finite
2651:square
2528:
2474:
2451:
2427:
2397:
2372:
2030:. The
1151:up to
1145:centre
835:where
711:
606:
541:
143:vector
42:, the
2370:S2CID
2302:Notes
2230:or a
1934:code
1467:union
986:then
894:Proof
68:space
2526:ISBN
2472:ISBN
2449:ISBN
2425:ISBN
2395:ISBN
2130:and
1041:ball
682:>
533:Let
124:word
34:and
2621:Set
2616:Bin
2558:doi
2522:GTM
2498:doi
2468:GTM
2362:doi
2034:of
1978:of
697:).
525:.)
30:In
2797::
2552:.
2546:.
2520:.
2494:24
2492:.
2488:.
2466:,
2411:;
2393:.
2389:.
2368:.
2356:.
2234:.
2160:=
2156:=
2148:≤
2134:≤
2126:≤
2054:.
86:.
2594:e
2587:t
2580:v
2566:.
2560::
2554:5
2534:.
2504:.
2500::
2457:.
2433:.
2403:.
2376:.
2364::
2358:5
2317:.
2255:t
2251:t
2243:t
2239:t
2220:q
2199:n
2194:q
2188:A
2162:t
2158:r
2154:s
2150:r
2146:s
2138:.
2136:r
2132:t
2128:t
2124:s
2100:)
2097:1
2091:d
2088:(
2083:2
2080:1
2069:=
2065:t
2048:C
2044:s
2040:s
2036:C
2028:C
2024:r
2008:n
2003:q
1997:A
1984:r
1980:C
1960:n
1955:q
1949:A
1936:C
1922:)
1919:d
1916:,
1913:n
1910:(
1905:q
1901:A
1862:.
1852:k
1848:)
1844:1
1838:q
1835:(
1829:)
1824:k
1821:n
1816:(
1808:t
1803:0
1800:=
1797:k
1784:n
1780:q
1771:)
1768:d
1765:,
1762:n
1759:(
1754:q
1750:A
1723:.
1718:n
1714:q
1701:k
1697:)
1693:1
1687:q
1684:(
1678:)
1673:k
1670:n
1665:(
1657:t
1652:0
1649:=
1646:k
1631:)
1628:d
1625:,
1622:n
1619:(
1614:q
1610:A
1606:=
1603:m
1597:)
1594:d
1591:,
1588:n
1585:(
1580:q
1576:A
1550:n
1546:q
1542:=
1538:|
1532:n
1527:q
1521:A
1514:|
1491:n
1486:q
1480:A
1453:t
1433:C
1413:)
1410:d
1407:,
1404:n
1401:(
1396:q
1392:A
1369:.
1360:k
1356:)
1352:1
1346:q
1343:(
1337:)
1332:k
1329:n
1324:(
1316:t
1311:0
1308:=
1305:k
1293:=
1290:m
1265:n
1260:q
1254:A
1231:q
1211:)
1208:1
1202:q
1199:(
1179:n
1159:t
1131:t
1111:m
1091:t
1071:c
1051:t
1027:C
1021:c
998:t
962:)
959:1
953:d
950:(
945:2
942:1
932:=
929:t
906:d
878:.
869:2
865:1
859:d
849:=
846:t
815:k
811:)
807:1
801:q
798:(
792:)
787:k
784:n
779:(
771:t
766:0
763:=
760:k
749:n
745:q
736:)
733:d
730:,
727:n
724:(
719:q
715:A
685:1
677:n
673:q
652:d
629:n
609:C
586:q
566:)
563:d
560:,
557:n
554:(
549:q
545:A
513:q
491:q
485:A
462:q
440:q
434:A
409:n
404:q
398:A
375:n
355:q
333:n
329:q
306:n
301:q
295:A
270:q
264:A
241:n
221:q
199:q
193:A
167:n
163:q
159:q
155:n
151:m
147:m
139:m
135:q
128:n
116:n
112:n
108:m
104:n
96:q
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.