Table of Nonlinear Binary Codes

Keywords: A(n,d), binary codes, bounds, clique-finding, nonlinear codes, perfect codes

A(n,d) is the size of the largest binary code of length n and minimal distance d (the most fundamental quantity in coding theory).
This file contains a table of lower bounds (and in some cases the exact values) of A(n,d).
We are also planning to include explicit codes for as many of these entries as we can locate.

Last updated November 24, 1999.


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

Maintained by

Simon Litsyn, Tel-Aviv University
Department of Electrical Engineering-Systems, Ramat-Aviv 69978, Tel-Aviv, Israel
Email address: litsyn@eng.tau.ac.il; Home page

E. M. Rains, AT&T Labs-Research
Room C290, 180 Park Ave., PO Box 971, Florham Park NJ 07932-0971 USA
Email address: rains@research.att.com; Home page

and

N. J. A. Sloane, AT&T Labs-Research
Room C233, 180 Park Ave., PO Box 971, Florham Park NJ 07932-0971 USA
Email address: njas@research.att.com; Home page

Notes

  1. Contributions of new records, explicit codes or upper bounds are welcomed. All contributions will be acknowledged.

  2. Format of Table

    Format of table is

    n       d       k       N       Type       Reference
    meaning that

    A(n,d) >= N*2^k.
    If n is missing, let the next entry be

    n0       d       k       N       Type       Reference
    then

    A(n,d) >= N*2^(k-(n0-n)).

  3. The table covers the range minimal distance d <= 29 and length n <= 512.

  4. The primary source for this table is Simon Litsyn's chapter "Tables of Best Known Binary Codes" which will soon be published in the Handbook of Coding Theory (edited V. Pless et al., North-Holland, 1998, to appear). That table was in turn based on earlier tables of F. J. MacWilliams and N. J. A. Sloane (see for example Appendix A of The Theory of Error-Correcting Codes, North-Holland, 1977).

  5. In the earlier tables the codes were described by giving the redundancy r, rather than the number of codewords M = N*2^k. These quantities are related by

    r   =   n   -   log_2(M)   =   n   -   k   -   log_2(N).

    At the end of this page we give a table to assist in converting between the values of r and M.

  6. The table only considers odd values of d, since (if d is odd) A(n, d) = A(n+1, d+1).

  7. There is a companion table of A(n,d,w), the size of the largest binary code of length n, minimal distance d and constant weight w.

  8. See also our home pages: SL | EMR | NJAS

Memo to algorithms specialists:

This file contains a large number of clique-finding problems. Construct the graph whose vertices represent binary strings of length n. Join two vertices by an edge if and only if the Hamming distance bewteen the strings is at least d. Then what we are interested in is the quantity A(n,d), the size of the largest clique in this graph.

This file contains a large number of lower bounds on this clique size. If you can improve any of these entries or establish the optimality of any entries that we don't already know are optimal (these are indicated by a period after the number) please let us know (send us the clique too!).



[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]



Start of Tables


TABLE d=3 Lower bounds on A(n,3)
ndkNTypeReference
7341HG[hamm50]
9335SW[best80]
11349SW[juli65]
153111HG[hamm50]
163585SW[roma83]
173683SW[etzi00]
183841MT[hama88,kaba88]
193125UV[best80]
233159UV[sloa70a]
313261HG[hamm50]
3332185UV[roma83]
3532383UV[etzi00]
3732641UV[hama88,kaba88]
393315UV[best80]
473389UV[sloa70a]
633571HG[hamm50]
703431657009MT[kaba88]
713633MT[oste96]
7536341UV[hama88,kaba88]
793705UV[best80]
953859UV[sloa70a]
12731201HG[hamm50]
14131131657009UV[kaba88]
14331343UV[oste96]
151313841UV[hama88,kaba88]
15931495UV[best80]
163315119MT[oste96]
19131809UV[sloa70a]
25532471HG[hamm50]
27032021021273028302258913MT[kaba88]
28332541657009UV[kaba88]
28732773UV[oste96]
30032203348824985082075276195MT[kaba88]
303328941UV[hama88,kaba88]
31933085UV[best80]
327331419UV[oste96]
38333719UV[sloa70a]
51135021HG[hamm50]
51234431021273028302258913MT[kaba88]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

TABLE d=5 Lower bounds on A(n,5)
ndkNTypeReference
8521LI[cord67]
11533HD[plot60]
15581PR[nord67]
195111XP[sema73,sloa72a]
20595XP[sloa72a]
235141LI[wagn66]
335231LI[chen91]
355225YP[krac83a]
3652033YG[zino85]
37519129YG[zino85]
635521PR[prep68]
705581XP[sema73,sloa72a]
815681LI[shea]
12851141B[sloa72a]
13551179YP[krac83a]
1365114133YG[zino85]
13751121057YG[zino85]
1385114525YG[zino85]
13951141043YG[zino85]
1405117259YG[zino85]
14151161029YG[zino85]
25552401PR[prep68]
27152551XP[sema73,sloa72a]
27852611GB[chie75]
51254941B[sloa72a]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

TABLE d=7 Lower bounds on A(n,7)
ndkNTypeReference
11721LI[cord67]
15751RM[mull54]
16729CM[sloa70a]
237121GO[gola49]
247121--
277141DC[karl69]
317171CS[chen89]
377221LI[shea]
397215YG[krac83a]
637471IM[goet74]
697521XG[krac83b]
7074911XG[zino82]
757571XH[krac83b]
7675411XH[zino82]
887691LI[shea]
957751LI[shea]
12871071GP[gopp70]
14371189YG[krac83a]
25572331IM[goet74]
26472411XG[kats87]
27372491XH[kats87]
27472491--
31172851SV[helg72]
51274851GP[gopp70]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

TABLE d=9 Lower bounds on A(n,9)
ndkNTypeReference
14921LI[cord67]
15921--
16913HD[leve61]
19935HD[leve61]
209121NL[kaik89]
219125NL[leve61]
229219NL[kaik97]
23971LI[hash76]
249411NL[kaik97]
2591135NL[kaik97]
279101CY[pire80]
2892365NL[kaik97]
319131LI[shea]
359153YC[sloa72a]
369153--
3791411Y1[zino84a]
389165Y1[zino84a]
399169Y1[zino84a]
419211QR[berl68]
429203Y1[zino84a]
439205Y1[zino84a]
459241QR[karl69]
469219Y1[lits98]
4792117Y1[lits98]
489261LI[shea]
4992315YC[sloa72a]
509257Y1[lits98]
5192513Y1[lits98]
529283Y1[lits98]
5392711Y1[lits98]
549295Y1[lits98]
559299Y1[lits98]
569331Y1[lits98]
579317Y1[lits98]
589333Y1[lits98]
599335Y1[lits98]
649401GP[macw77]
719461XB[bier97]
769501XQ[kasa75]
779479XQ[zino84b]
919641AL[helg74]
12891001B[sloa72a]
13591061XB[sloa72b]
136910093XB[zino82]
14291121XQ[kasa75]
14391099XQ[zino84b]
16791361AL[helg74]
25692241B[sloa72a]
26592321GP[sugi76]
27592411XB[zino82]
31592801YB[cace91,rodi]
51294761B[sloa72a]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

TABLE d=11 Lower bounds on A(n,11)
ndkNTypeReference
171121LI[cord67]
181121--
191113HD[leve61]
201131B[gold68]
231143HD[leve61]
2411213CM[sloa70b]
261171HS[helg73a]
2711189NL[kaik97]
3111111B[bose60,hoqu59]
3311121LI[shea]
3611141LI[mori93]
4711241QR[karl69]
4911241--
6311371Z4[cald94,cald96]
6411371--
6711391XB[sloa72b]
6811391--
6911383XB[zino82]
7111411ZV[zino76]
7411431XB[tolh86]
7511423XB[zino82]
7811461XB[zino82]
8911561CY[prom78]
9911651GP[roel82]
12811931GP[gopp70]
13511991XB[sloa72b]
13611969XB[zino82]
142111051XQ[kasa75]
143119993XQ[zino84b]
149111111GP[kasa75]
150111089GP[zino84b]
191111521YB[helg73b]
256112161GP[gopp70]
264112231XB[sloa72b]
26511216151XB[zino82]
272112301XQ[kasa75]
27311223151XQ[zino84b]
280112371GP[sugi76]
28111230151GP[zino84b]
311112671SV[helg72]
512114671GP[gopp70]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

TABLE d=13 Lower bounds on A(n,13)
ndkNTypeReference
201321LI[cord67]
211321--
221321--
231313HD[leve61]
241331HD[leve61]
271337HD[plot60]
2813215CM[sloa70b]
291361RM[mull54]
321381LI[hell89]
341391CY[pire74]
3813121LI[bier95]
3913121--
4313151CY[berl68]
4513161XC[doug91]
4713171CY[jens85]
6313321Z4[ples95,cald96]
6413321--
6513321--
6613307Y1[zino84a]
6713323Y1[zino84a]
6813325Y1[zino84a]
7013361XB[sloa72b]
7313381XQ[kasa75]
7413381--
7513373XQ[zino84b]
7713401QR[karl69]
7913411ZV[kasa76]
8113413XZ[zino84b]
9013511CY[prom78]
9413541LI[brou93]
10113601GP[roel82]
12813861GP[gopp70]
13513921XB[sloa72b]
13613899XB[zino82]
14213981XQ[kasa75]
14313959XQ[zino84b]
149131041GP[sugi76]
150131019GP[zino84b]
156131101GP[sugi76]
157131079GP[zino84b]
191131441YB[helg73b]
256132081B[sloa72a]
264132151XB[sloa72b]
26513208151XB[zino82]
272132221XQ[kasa75]
27313215151XQ[zino84b]
280132291GP[sugi76]
28113222151GP[zino84b]
288132361GP[sugi76]
28913229151GP[zino84b]
327132741YB[helg73b]
512134581B[dels75]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

TABLE d=15 Lower bounds on A(n,15)
ndkNTypeReference
231521LI[cord67]
251521--
261513HD[leve61]
271531HD[leve61]
281515HD[leve61]
311561RM[mull54]
321561--
331539HD[leve61]
351581LI[dodu87]
371591LI[fark94]
3915101ZV[sugi76,zino76]
401585XB[zino82]
4415141LI[bouk95]
4515141--
4815161LI[fark94]
5015171CY[berl68]
5515211CY[berl68]
6315281CY[kasa69]
7215361CY[prom78]
7315361--
7415361--
7915401QR[karl69]
8015401--
8115379Y1[zino84a]
8215411Y1[zino84a]
8315397Y1[zino84a]
8415413Y1[zino84a]
8515415Y1[zino84a]
8715451QR[karl69]
8815451--
8915451--
9015451--
91154211Y1[zino84a]
9215445Y1[zino84a]
9315449Y1[zino84a]
9415481Y1[zino84a]
951540497YB[zino85]
9615483Y1[zino84a]
9715485Y1[zino84a]
9915521QR[karl69]
1001544379Y2[zino84a]
101155011Y2[zino84a]
1021547163Y2[zino84a]
1031547301Y2[zino84a]
1041548277Y2[zino84a]
10515571GP[roel82]
106155329Y2[zino84a]
1071551211Y2[zino84a]
1081552191Y2[zino84a]
109155543Y2[zino84a]
110155577Y2[zino84a]
1111555137Y2[zino84a]
12815791GP[gopp70]
13515851XB[sloa72b]
13615829XB[zino82]
14215911XQ[kasa75]
14315889XQ[zino84b]
14915971GP[sugi76]
15015949GP[zino84b]
156151031GP[sugi76]
157151009GP[zino84b]
163151091GP[sugi76]
164151069GP[zino84b]
191151361YB[helg73b]
256152001GP[gopp70]
264152071XB[sloa72b]
26515200151XB[zino82]
272152141XQ[kasa75]
27315207151XQ[zino84b]
280152211GP[sugi76]
28115214151GP[zino84b]
288152281GP[sugi76]
28915221151GP[zino84b]
296152351GP[sugi76]
29715228151GP[zino84b]
327152651YB[helg73b]
512154491GP[gopp70]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

TABLE d=17 Lower bounds on A(n,17)
ndkNTypeReference
261721LI[cord67]
271721--
281721--
291713HD[leve61]
301713--
311731HD[leve61]
321723--
351739HD[plot60]
3617219CM[sloa70b]
371745HD[leve61]
3817221CM[sloa70b]
391771LI[tilb81]
411781GP[allt77]
4417101CY[gull94]
4617111YQ[gron94]
5017125YQ[lits98]
5117125--
5317161GP[loel84]
5417161--
5617171LI[mori94]
6017201CY[weij97]
6517241LI[gron92]
6617241--
6717233Y1[zino84a]
6817235Y1[zino84a]
7117281YC[sloa72a]
8917451QR[karl69]
9017451--
9117451--
9217451--
93174211Y1[zino84a]
9417445Y1[zino84a]
9517449Y1[zino84a]
9617481Y1[zino84a]
9717467Y1[zino84a]
9817483Y1[zino84a]
9917485Y1[zino84a]
10117521QR[karl69]
10217521--
10317505Y1[zino84a]
10517541DC[karl69]
106175111Y1[zino84a]
107175121Y1[zino84a]
10817545Y1[zino84a]
109175319Y1[zino84a]
11017559Y1[zino84a]
111175517Y1[zino84a]
11217601Y1[zino84a]
113175715Y1[zino84a]
11417597Y1[zino84a]
115175913Y1[zino84a]
11617623Y1[zino84a]
117176111Y1[zino84a]
11817635Y1[zino84a]
11917639Y1[zino84a]
12017671Y1[zino84a]
12117657Y1[zino84a]
12217673Y1[zino84a]
12317675Y1[zino84a]
12517711B[bose60,hoqu59]
12617711--
12817721B[sloa72a]
13517781XB[sloa72b]
13617759XB[zino82]
14217841XQ[kasa75]
14317819XQ[zino84b]
14917901GP[sugi76]
15017879GP[zino84b]
15617961GP[sugi76]
15717939GP[zino84b]
163171021GP[sugi76]
16417999GP[zino84b]
170171081GP[sugi76]
171171059GP[zino84b]
191171281YB[helg73b]
256171921B[sloa72a]
264171991XB[sloa72b]
26517192151XB[zino82]
272172061XQ[kasa75]
27317199151XQ[zino84b]
280172131GP[sugi76]
28117206151GP[zino84b]
288172201GP[sugi76]
28917213151GP[zino84b]
296172271GP[sugi76]
29717220151GP[zino84b]
304172341GP[sugi76]
30517227151GP[zino84b]
331172601YB[helg73b]
512174401B[sloa72a]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

TABLE d=19 Lower bounds on A(n,19)
ndkNTypeReference
291921LI[cord67]
321931--
331913HD[leve61]
341931CY[chen70]
351915HD[leve61]
361923HD[leve61]
391945HD[plot60]
4019221CM[sloa70b]
4119311HD[leve61]
421971LI[tilb81]
441981AX[allt77]
4719101CY[gull97]
4919111YQ[gron94]
5519161GP[loel84]
5619161--
5719161--
5919171DG[dels75]
6119181B[bose60,hoqu59]
6519211LI[zhi93]
7219271CY[prom78]
7319271--
7519281LI[brou93a]
7619281--
8019311YQ[gron94]
8619361YQ[gron94]
10319521QR[karl69]
10419521--
10719541QR[karl69]
10819541--
10919541--
12719711B[bose60,hoqu59]
12819711--
12919711--
13119721GP[sugi76]
13519751GP[sugi76]
13719745XB[zino82]
13819749XB[zino82]
13919781XC[sloa72a]
14019781--
141197269XB[kats87]
14219791XB[bier97]
14419801XB[bier97]
14519779XB[zino82]
14719821NL[sugi76]
14819805XB[zino82]
15919931XB[zino84a]
191191241YB[helg73b]
255191871B[bose60,hoqu59]
260191911XB[sloa72b]
261191911--
262191903XB[zino82]
263191921XB[zino82]
2641918911XB[kats87]
267191929XB[zino82]
271191991XB[zino82]
273191991--
280192051GP[sugi76]
28119198151GP[zino84b]
288192121GP[sugi76]
28919205151GP[zino84b]
296192191GP[sugi76]
29719212151GP[zino84b]
304192261GP[sugi76]
30519219151GP[zino84b]
319192401GP[zino84a]
383193031GP[zino84a]
512194311GP[gopp70]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

TABLE d=21 Lower bounds on A(n,21)
ndkNTypeReference
322121LI[cord67]
352131--
362113HD[leve61]
372113--
382131HD[leve61]
392115HD[leve61]
402117HD[leve61]
4321311HD[plot60]
4421311--
452153HD[leve61]
472181GP[allt76]
482181--
492181--
5021513YD[zino85]
512183YD[zino85]
5221711YD[zino85]
532195YD[zino85]
542199YD[zino85]
5521131YD[zino85]
56211011YD[zino85]
5721943YD[zino85]
5821133Y1[zino84a]
5921135Y1[zino84a]
6121171YD[dels75]
6321181B[bose60,hoqu59]
6421181--
6521181--
6621181--
6821191ZV[wise83]
7121211ZV[wise83]
7421231ZV[wise83]
7721251ZV[wise83]
8021271ZV[wise83]
8321291CY[jens85]
8621311B[hart73]
8721311--
8821311--
9021321LI[kupp95]
9221331CY[prom78]
9521351ZV[zino76]
10021391XZ[chen87]
101213611ZV[zino76]
10521431YB[helg73b]
12721641B[bose60,hoqu59]
13521711XB[sloa72b]
13621711--
137216569XB[zino82]
138216723XB[zino82]
13921721XB[zino82]
14221741GP[sugi76]
14421735XB[zino82]
14521739XB[zino82]
14621739--
148217319XB[zino82]
14921781XB[zino82]
15021781--
15121765GP[zino84b]
15221769GP[zino84b]
15421811NL[sugi76]
15521795GP[zino84b]
15821841XQ[kasa75]
191211161YB[helg73b]
255211791B[bose60,hoqu59]
264211871XB[sloa72b]
26521180151XB[zino82]
268211901XQ[kasa75]
269211901--
270211893GP[zino84b]
271211911XQ[krac83b]
2722118811XB[kats87]
275211919GP[zino84b]
279211981GP[zino84b]
280211981--
281211981--
288212041GP[sugi76]
28921197151GP[zino84b]
296212111GP[sugi76]
29721204151GP[zino84b]
304212181GP[sugi76]
30521211151GP[zino84b]
312212251GP[sugi76]
31321218151GP[zino84b]
320212321GP[sugi76]
32121225151GP[zino84b]
383212941GP[zino84a]
512214221B[sloa72a]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

TABLE d=23 Lower bounds on A(n,23)
ndkNTypeReference
352321LI[cord67]
362321--
372321--
382321--
392313HD[leve61]
402313--
412331HD[leve61]
422331--
432323HD[leve61]
442341CY[chen70]
472353HD[leve61]
4823225CM[sloa70b]
502381CY[chen70]
512381--
522381--
542383YD[zino85]
5523101ZV[zino76]
572397YD[zino85]
6323171DG[dels75]
6423171--
6623181XQ[sloa72b]
6923191--
7123191ZV[wise83]
7423211ZV[wise83]
7723231ZV[wise83]
8023251ZV[wise83]
8323271ZV[wise83]
8723301LI[karl69]
8923311ZV[wise83]
9023311--
9123311--
9223311--
9523331ZV[zino76]
9823351ZV[tolh86]
10123371LI[brou93]
10423391ZV[brou97]
10723411ZV[chen87]
11623491CY[scho92]
11723491--
11923501ZV[chen87]
12723571B[bose60,hoqu59]
13523641XB[sloa72b]
13623619XB[zino82]
14223701XQ[kasa75]
14323701--
144236469GP[zino84b]
145236715XB[zino82]
14623711XB[zino82]
14923731GP[sugi76]
15123725GP[zino84b]
15223729GP[zino84b]
15323761GP[sugi76]
15423761--
155237219GP[zino84b]
156237219--
16123811ZV[kasa76]
162237717XZ[zino82]
16723861ZV[kasa76]
168238217XZ[zino82]
17323911ZV[kasa76]
174238717XZ[zino82]
207231241YB[helg73b]
255231711B[bose60,hoqu59]
264231791XB[sloa72b]
26523172151XB[zino82]
272231861XQ[kasa75]
27323179151XZ[zino84b]
276231891GP[sugi76]
277231891--
278231883GP[zino84b]
279231901GP[zino84b]
2802318711XB[kats87]
283231909GP[zino84b]
287231971GP[zino84b]
288231971--
289231971--
296232031GP[sugi76]
29723196151GP[zino84b]
304232101GP[sugi76]
30523203151GP[zino84b]
312232171GP[sugi76]
31323210151GP[zino84b]
320232241GP[sugi76]
32123217151GP[zino84b]
328232311GP[sugi76]
32923224151GP[zino84b]
383232851YB[zino84a]
512234131GP[gopp70]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

TABLE d=25 Lower bounds on A(n,25)
ndkNTypeReference
382521LI[cord67]
392521--
402521--
412521--
422521--
432513HD[leve61]
442513--
452531HD[leve61]
462515HD[leve61]
472523HD[leve61]
482541HD[leve61]
5125313HD[plot60]
5225227CM[sloa70b]
532547HD[leve61]
552581YK[zino85]
5625347YK[zino85]
57252187YK[zino85]
582583Y1[zino84a]
592585Y1[zino84a]
6125121DG[dels75]
6325121--
6525131CY[chen70]
6725125Y1[zino84a]
6925161YD[zino82]
7025145YD[zino82]
7125171DG[dels75]
7225171--
7425181XQ[allt84]
7525181--
7625181--
7725181--
8025201CY[chen94]
8225211CY[chen94]
8425221LI[gron92]
8625231YB[helg73b]
8725231--
8925241LI[mori94]
9125251LI[gron92]
9225251--
9525271XZ[barg87]
9825291LI[gron92]
9925291--
10225311LI[gron92]
11125391YB[zino85]
11225391--
113253615Y1[zino84a]
11425387Y1[zino84a]
115253813Y1[zino84a]
11625413Y1[zino84a]
117254011Y1[zino84a]
11825425Y1[zino84a]
11925429Y1[zino84a]
12025461Y1[zino84a]
12125447Y1[zino84a]
12225463Y1[zino84a]
12325465Y1[zino84a]
12525501B[bose60,hoqu59]
12625501--
12725501--
1292512047781292273887XZ[barg87]
13525571XB[sloa72b]
13625549XB[zino82]
14225631XQ[kasa75]
14925691GP[sugi76]
15025691--
151256369GP[zino84b]
152256615GP[zino84b]
15325701GP[krac83b]
15625721GP[sugi76]
15825715GP[zino84b]
15925719GP[zino84b]
16125761ZV[kasa76]
162257217XZ[zino82]
16725811ZV[kasa76]
168257717XZ[zino82]
17325861ZV[kasa76]
174258217XZ[zino82]
17925911ZV[kasa76]
180258717XZ[zino82]
18525961ZV[kasa76]
186259217XZ[zino82]
191251011ZV[kasa76]
192259717XZ[zino82]
207251161YB[helg73b]
255251631B[bose60,hoqu59]
264251711XB[sloa72b]
26525164151XB[zino82]
272251781XQ[kasa75]
27325171151XQ[zino84b]
280251851GP[sugi76]
28125178151GP[zino84b]
284251881GP[sugi76]
285251881--
286251873GP[zino84b]
287251891GP[zino84b]
2882518611XB[kats87]
291251899GP[zino84b]
295251961GP[zino84b]
297251961--
304252021GP[sugi76]
30525195151GP[zino84b]
312252091GP[sugi76]
31325202151GP[zino84b]
320252161GP[sugi76]
32125209151GP[zino84b]
328252231GP[sugi76]
32925216151GP[zino84b]
336252301GP[sugi76]
33725223151GP[zino84b]
383252761YB[zino84a]
512254041B[sloa72a]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

TABLE d=27 Lower bounds on A(n,27)
ndkNTypeReference
412721LI[cord67]
422721--
432721--
442721--
452721--
462713HD[leve61]
472713--
482731CY[chen70]
492731--
502715HD[leve61]
512717HD[leve61]
522719HD[leve61]
552747HD[plot60]
562747--
5727231YK[zino85]
6327121DG[dels75]
6427121--
6527121--
6727131XD[zino82]
7127161XD[zino82]
7227145XD[zino82]
7327171DG[macw77]
7427171--
7527171--
7627171--
7827181XQ[allt84]
7927181--
8027181--
8427211CY[weij97]
8527211--
8827231B[helg73b]
8927231--
9127241Z4[aydi99]
9227241--
9327241--
9527251LI[gron92]
9627251--
9827261LI[gron92]
99272077Y2[lits98]
1002720137Y2[lits98]
1012721121Y2[lits98]
102272353Y2[lits98]
103272523Y2[lits98]
104272479Y2[lits98]
107272837YC[lits98]
108272837--
109272929Y2[lits98]
110273111Y2[lits98]
11227361CY[scho92]
12727501B[bose60,hoqu59]
12827501--
12927501--
13127511GP[sugi76]
13527541GP[sugi76]
13727535XB[zino82]
13827539XB[zino82]
13927571XC[sloa72a]
14027571--
141275169XB[kats87]
14227581XB[bier97]
14427591XB[bier97]
14727611NL[sugi76]
14827595XB[zino82]
15127641GP[sugi76]
15627681GP[sugi76]
15727681--
158276269GP[zino84b]
15927691XZ[barg87]
16727761ZV[kasa76]
168277217XZ[zino82]
17327811ZV[kasa76]
174277717XZ[zino82]
17927861ZV[kasa76]
180278217XZ[zino82]
18527911ZV[kasa76]
186278717XZ[zino82]
19127961ZV[kasa76]
192279217XZ[zino82]
197271011ZV[kasa76]
198279717XZ[zino82]
199271003XZ[zino82]
200271021XZ[zino82]
201279745XZ[kats87]
204271029XZ[zino82]
207271081YB[helg73b]
255271551B[bose60,hoqu59]
264271631XB[sloa72b]
26527156151XB[zino82]
272271701XQ[kasa75]
27327163151XQ[zino84b]
280271771GP[sugi76]
28127170151GP[zino84b]
288271841GP[sugi76]
28927177151GP[zino84b]
292271871GP[sugi76]
293271871--
294271863GP[zino84b]
295271881GP[zino84b]
2962718511XB[kats87]
299271889GP[zino84b]
303271951GP[zino84b]
304271951--
305271951--
312272011GP[sugi76]
31327194151GP[zino84b]
320272081GP[sugi76]
32127201151GP[zino84b]
328272151GP[sugi76]
32927208151GP[zino84b]
336272221GP[sugi76]
33727215151GP[zino84b]
344272291GP[sugi76]
34527222151GP[zino84b]
383272671GP[sugi76]
512273951GP[gopp70]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

TABLE d=29 Lower bounds on A(n,29)
ndkNTypeReference
442921LI[cord67]
452921--
462921--
472921--
482921--
492913HD[leve61]
502913--
512913--
522931HD[leve61]
532915HD[leve61]
542923HD[leve61]
552917HD[leve61]
562925HD[leve61]
5929315HD[plot60]
6029231CM[sloa70b]
612971RM[mull54]
622971--
6329317HD[leve61]
642981LI[dodu87]
6929121XD[macw77]
7029121--
7129121--
7329131XD[zino82]
7429131--
7529115YZ[zino85]
7829161LI[gron92]
8029171XD[zino82]
8129171--
8229171--
8329171--
8529181XB[zino82]
8629181--
8729181--
9329231LI[gron94]
9429231--
9529231--
9629231--
9829241LI[gron94]
10029251CY[weij97]
10729289YC[lits98]
10829289--
10929321Y1[lits98]
11029307Y1[lits98]
11129323Y1[lits98]
11229325Y1[lits98]
11429361CY[scho92]
11529361--
11729371CY[scho92]
11829355Y1[zino84a]
11929359Y1[zino84a]
12029391Y1[zino84a]
12129377Y1[zino84a]
12229393Y1[zino84a]
12329395Y1[zino84a]
12529431B[kasa69]
12629431--
12729431--
13529501XB[sloa72b]
13629501--
137294469XB[zino82]
138294623XB[zino82]
13929511XB[zino82]
14229531GP[sugi76]
14429525XQ[zino82]
14529529XQ[zino82]
14629561XQ[kasa75]
14729561--
148295219XB[zino82]
14929571XB[zino82]
15429611XZ[barg87]
155295717XZ[barg87]
15929651XZ[barg87]
160296117XZ[barg87]
16429691XZ[barg87]
165296517XZ[barg87]
16629683XZ[barg87]
16729701ZV[kasa76]
168296711XZ[kats87]
170296733XZ[barg87]
17929811ZV[kasa76]
180297717XZ[zino82]
18529861ZV[kasa76]
186298217XZ[zino82]
19129911ZV[kasa76]
192298717XZ[zino82]
19729961ZV[kasa76]
198299217XZ[zino82]
199299229XZ[zino84b]
2002990155Y1[shek89]
20129864589Y1[shek89]
20229874237Y1[shek89]
20329877807Y1[shek89]
2042991897Y1[shek89]
20529911645Y1[shek89]
20629913009Y1[shek89]
20729915489Y1[shek89]
208299939Y1[shek89]
20929942263Y1[shek89]
21029952045Y1[shek89]
21129953683Y1[shek89]
2122999413Y1[shek89]
21329100369Y1[shek89]
21429991313Y1[shek89]
21529992325Y1[shek89]
216291111Y1[shek89]
21729102897Y1[shek89]
21829103781Y1[shek89]
219291031351Y1[shek89]
22029107145Y1[shek89]
22129107247Y1[shek89]
22229107417Y1[shek89]
22329107697Y1[shek89]
224291149Y1[shek89]
22529110235Y1[shek89]
22629111189Y1[shek89]
22729111299Y1[shek89]
255291471B[bose60,hoqu59]
264291551XB[sloa72b]
26529148151XB[zino82]
272291621XQ[kasa75]
27329155151XQ[zino84b]
280291691GP[sugi76]
28129162151GP[zino84b]
288291761GP[sugi76]
28929169151GP[zino84b]
296291831GP[sugi76]
29729176151GP[zino84b]
300291861GP[sugi76]
301291861--
302291853GP[zino84b]
303291871GP[zino84b]
3042918411XB[kats87]
307291879GP[zino84b]
311291941GP[zino84b]
313291941--
320292001GP[sugi76]
32129193151GP[zino84b]
328292071GP[sugi76]
32929200151GP[zino84b]
336292141GP[sugi76]
33729207151GP[zino84b]
344292211GP[sugi76]
34529214151GP[zino84b]
352292281GP[sugi76]
35329221151GP[zino84b]
383292581GP[zino84a]
512293861B[sloa72a]
ndkNTypeReference


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]


End of Tables


KEY

AL = alternant code (linear)
AX = the (a+x, b+x, a+b+x) construction
B = BCH code
BV = Belov et al.'s linear codes meeting the Griesmer bound
CM = conference-matrix code
CS = code from a group
CY = cyclic or quasicyclic code
DC = double circulant code
DG = Delsarte-Goethals code
GB = generalized BCH code
GO = Golay code
GP = Goppa or lengthened Goppa code
HD = Hadamard matrix code
HG = Hamming code
HS = Helgert and Stinaff construction A (Griesmer shortening)
IM = Goethals nonlinear code I(M)
KS = Kasahara et al. generalized concatenated codes (linear or nonlinear)
LI = linear code (a generating or parity-check matrix is explicitly found)
MT = matrix construction
NL = nonlinear code (the list of codewords is explicitly given)
PR = Preparata code
PT = Piret's construction
QR = quadratic residue code
RM = Reed-Muller code
SV = Srivastava code
SW = nonlinear single error-correcting code
UV = the ({\bf u},{\bf u}+{\bf v})-construction
X = lengthening construction X
XB = lengthening of BCH codes
XC = lengthening of a cyclic code
XD = lengthening of Delsarte-Goethals codes
XG = lengthening of Preparata-Goethals codes
XH = lengthening of Hamming-Preparata-Goethals codes
XP = lengthening of Hamming-Preparata codes
XQ = two-dimensional lengthening of BCH codes
XZ = lengthening of generalized concatenated codes
X3 = construction X3
X4 = construction X4
Y1 = shortening of codes with j=1
Y2 = shortening of codes with j=2
YB = shortening of BCH codes
YC = shortening of cyclic or quadratic residue codes
YD = shortening of Delsarte-Goethals codes
YG = shortening of Goethals codes
YK = shortening of Kerdock codes
YP = shortening of Preparata codes
YQ = shortening of Goppa codes
YZ = shortening of generalized concatenated codes
ZV = generalized concatenated codes
Z4 = codes derived from Z_4 cyclic codes.


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]

Table of size, rate, redundancy.

For the values of N occuring in the main tables, the following table gives the contribution to the rate, i.e. log_2(N), and the corresponding fractional part of the redundancy, i.e. red = ceiling(log_2(N)) - log_2(N).

To illustrate the use of this table, consider the second line in the d=3 table, which describes a code of length n = 9, minimal distance d = 3, with k = 3, N = 5, hence has M = 2^3 * 5 = 40 codewords. Referring to the third line (N=5) of the table below, we find that this code has

rate = log_2(M) = k + log_2(N) = 3 + 2.32193 = 5.32193,

and

redundancy = n - log_2(M) = 9 - 6 + 0.67807 = 3.67807.

Table of size, rate, redundancy:
Nlog_2(N)red
1 0.00000 1.00000
3 1.58496 0.41504
5 2.32193 0.67807
7 2.80735 0.19265
9 3.16993 0.83007
11 3.45943 0.54057
13 3.70044 0.29956
15 3.90689 0.09311
17 4.08746 0.91254
19 4.24793 0.75207
21 4.39232 0.60768
23 4.52356 0.47644
25 4.64386 0.35614
27 4.75489 0.24511
29 4.85798 0.14202
31 4.95420 0.04580
33 5.04439 0.95561
35 5.12928 0.87072
37 5.20945 0.79055
39 5.28540 0.71460
41 5.35755 0.64245
43 5.42626 0.57374
45 5.49185 0.50815
47 5.55459 0.44541
49 5.61471 0.38529
51 5.67243 0.32757
53 5.72792 0.27208
55 5.78136 0.21864
57 5.83289 0.16711
59 5.88264 0.11736
63 5.97728 0.02272
65 6.02237 0.97763
67 6.06609 0.93391
69 6.10852 0.89148
71 6.14975 0.85025
73 6.18982 0.81018
75 6.22882 0.77118
77 6.26679 0.73321
79 6.30378 0.69622
81 6.33985 0.66015
83 6.37504 0.62496
85 6.40939 0.59061
87 6.44294 0.55706
89 6.47573 0.52427
91 6.50779 0.49221
93 6.53916 0.46084
95 6.56986 0.43014
97 6.59991 0.40009
99 6.62936 0.37064
101 6.65821 0.34179
103 6.68650 0.31350
105 6.71425 0.28575
107 6.74147 0.25853
109 6.76818 0.23182
111 6.79442 0.20558
113 6.82018 0.17982
115 6.84549 0.15451
117 6.87036 0.12964
119 6.89482 0.10518
121 6.91886 0.08114
123 6.94251 0.05749
125 6.96578 0.03422
127 6.98868 0.01132
129 7.01123 0.98877
133 7.05528 0.94472
135 7.07682 0.92318
137 7.09803 0.90197
145 7.17991 0.82009
151 7.23840 0.76160
155 7.27612 0.72388
163 7.34873 0.65127
187 7.54689 0.45311
189 7.56224 0.43776
191 7.57743 0.42257
211 7.72110 0.27890
235 7.87652 0.12348
247 7.94837 0.05163
259 8.01681 0.98319
277 8.11374 0.88626
299 8.22400 0.77600
301 8.23362 0.76638
365 8.51175 0.48825
369 8.52748 0.47252
379 8.56605 0.43395
413 8.69000 0.31000
417 8.70390 0.29610
497 8.95710 0.04290
525 9.03617 0.96383
697 9.44501 0.55499
781 9.60918 0.39082
897 9.80896 0.19104
1029 10.00703 0.99297
1043 10.02652 0.97348
1057 10.04576 0.95424
1313 10.35865 0.64135
1351 10.39981 0.60019
1645 10.68387 0.31613
2045 10.99789 0.00211
2263 11.14402 0.85598
2325 11.18302 0.81698
3009 11.55507 0.44493
3683 11.84667 0.15333
4237 12.04883 0.95117
4589 12.16396 0.83604
5489 12.42233 0.57767
7807 12.93055 0.06945
1657009 20.66015 0.33985
2047781292273887 51.86298 0.13702
1021273028302258913 59.82507 0.17493
3348824985082075276195 71.50414 0.49586


[ Go to: distance 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29; KEY; References ]



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