Bounds for constant-weight codes

These tables give, for all parameters n, d, and w, an interval for the maximum size A(n,d,w) of a constant-weight binary code. A single value indicates that the lower and upper bounds coincide: A(n,d,w) is known exactly in such cases. Superscripts denote the methods by which the upper bounds were obtained, as detailed in the legend below.

The tables give the best known bounds on A(n,d,w) for all n up to 28 and all even d up to 14. For each n and d, w ranges from d/2+1 to the integer part of n/2. The values of A(n,d,w) for w outside this interval or for odd d are given in [1, Theorem 8]. In particular, bounds for w > n/2 follow from A(n,d,w) = A(n,d,n-w). For d = 16 or 18, exact values of A(n,d,w) are given in [5].

A corresponding resource for lower bounds of constant-weight codes is maintained by E. M. Rains and N. J. A. Sloane at [RS], an online version of [5]. The lower bounds in the tables below were copied from [RS] in December 1999 and are not regularly updated. See [RS] for updated lower bounds and details on how they were obtained. Some improved lower bounds, and extensions to larger values of n, were given in [9] in February 2002.

Table history

Please report further updates or corrections.

Legend of superscripts

Bounds on A(n,4,w)

w=345678910111213 14
n=645n=6
7 757
885 1458
9129 1899
10139 30936910
11 17935966911
1220951980O 132912
13269 659123- 129O166- 171O 13
14289919169- 1829 278- 2999325- 342914
15 3591059237- 27113389- 4559585- 640915
16 3791409315- 3369615- 7229836- 104091170- 1280916
17449156- 1579441- 474O854- 95291416- 175391770- 2210917
18489 1989518- 56591260- 142292041- 244893186- 394493540- 4420918
195792289692- 75291620- 178993172- 385994667- 581496726- 8326919
20 6092859874- 91292304- 250694213- 511197730- 9647910039- 12920913452- 16652920
21 70931591071- 119792856- 319296156- 7518910753- 134169 16897- 22509920188- 27132921
227393859138693927- 438998252- 10032916430- 206749 25570- 32794936381- 49519939688- 54264922
23839418- 4199177195313911638- 14421923276- 28842940786- 528339 57436- 75426973794- 103539923
2488949891895- 20119 7084915656- 18216934914- 432639 59387- 76912996496- 1267999116937- 1645659146552- 207078924
25 100955092334- 249097772- 8379921106- 25300946872- 569259 88748- 1201759140605- 1922809196449- 2881799228901- 342843925
26 104965092670- 2860910010- 10790926920- 31122965364- 822259 128050- 1644509218905- 3124559315700- 4544809398381- 6243879425950- 6856869 26
27117970293276- 3510912012- 12870935510- 416189 87709- 1050369186058- 2466759330347- 4440159510571- 7669359675262- 10225809 778872- 1296803927
281219 81993718- 3931915288- 163809 44747- 514809121403- 1456639260224- 3267789502068- 6906909806303- 11302209 1154541- 178951591400118- 220248091520224- 2593606928
w=345678910 11121314


Bounds on A(n,6,w)

w=4567891011121314
n=8210n=8
9359
105565 10
116511511
129512522512
131351821269 13
1414528204220 422114
15155 429702069O15
162094891129109- 1229120- 138916
17 2021689112- 124O166- 2079184- 259917
18 22969- 729132- 1869243- 3189260- 42820304- 42520 18
19252176- 839172- 2289338- 5039408- 718T504- 7892019
2030984- 1009232- 2769462- 6519588- 110714832- 136320944- 14212020
21319108- 1269269- 3509570- 8289774- 1695141184- 2359T1454- 2685T21
22 379132- 1369319- 4629759- 110091139- 227791792- 3736T2182- 4415T2636- 50642022
23 409147- 1709399- 5219969- 151891436- 316292271- 581992970- 7521203585- 79532023
24 429168- 1929532- 68091368- 178691882- 455493041- 843294200- 12186145267- 1468295616- 159062024
255092109700- 80091900- 242892590- 558194127- 12620146036- 19037147960- 2463020 9031- 30587925
26529 260991092600- 297193532- 789195703- 1612298695- 2889314 12037- 42075T14836- 50169T15977- 61174926
27549260- 280911709351094786- 1002797727- 23673912368- 435299 18096- 660792023879- 845742027553- 910802027
28639280- 30291170- 13069468096315- 12285910313- 31195917447- 6375614 29484- 1042312040188- 1421171449462- 1642202052995- 1697402028
w=45 67891011121314


Bounds on A(n,8,w)

w=567891011121314
n=1025n=10
1121011
123545 12
1331041013
14410758514
1565105155 15
16610165165 30516
17710 175242134917
1891021933O46- 49O48- 58O18
19 125289529789 88- 103919
20165 4098091309160- 1739176- 206920
21 21556912092109 280- 3029336- 363921
22 212177917693309 280- 473T616- 634T672- 680T22
2323577- 809 25395069400- 707T616- 1025T1288T23
24 24578- 929253- 2749 7599640- 1041T960- 1551T1288- 2142T25762024
25 3091009254- 3289759- 8569829- 1486T1248- 2333T1662- 3422T2576- 4087T25
26 30211309257- 3719760- 10669883- 2108T1519- 3496T1988- 5225T3070- 6741T3588- 7080T26
2731- 329130- 1359 278- 5009766- 12529970- 291414 1597- 4986T2295- 7833T3335- 10547204094- 11981T27
28 339130- 1499296- 5409833- 175091107- 389591820- 7016T2756- 11939144916- 17011T4805- 21152T 6090- 22710T28
w=5678910 11121314


Bounds on A(n,10,w)

w=67891011121314
n=12 25n=12
1325 13
1421021014
15353515
16 3104541016
17310556517
184106595105 18
1941085 121019519
20 51010101721205 38520
2175 131121527- 35938- 42921
2275 162124- 33935- 51946- 72T46- 80T22
23 85202133- 46945- 812054- 117965- 13520 23
2491024538- 609 56- 118T72- 1712095- 22320122- 2472024
25101028- 32948- 75972- 15820100- 26220125- 380T132- 434T 25
2613528- 361454- 1049 91- 21420130- 406T168- 566T 195- 702T210- 754T26
27 141036- 481466- 1219118- 29920162- 571T222- 882T351- 1201T405- 1419T27
28 161037- 56978- 1689132- 3769210- 82120286- 1356T365- 1977T756- 243820790- 26292028
w=67891011121314


Bounds on A(n,12,w)

w=7891011121314
n=14 25n=14
1525 15
162521016
1721021017
18 35354518
19353104519
20310555565 20
2131055 757521
22 456585115 12522
23410 6101010161023523
2441095165 24524546524
2551010525528- 37T36- 42950925
2651013526533- 482039- 66T54- 832158- 91T26
27610 151039939- 64T54- 1002182- 1402086- 15620 27
2885191139- 4520 49- 87T65- 1492084- 1992099- 24520172- 2652028
w=78910 11121314


Bounds on A(n,14,w)

w=891011121314
n=16 25n=16
1725 17
18252518
19 2521019
20 21021021020
2135353521
22353104545 22
2335310 41041023
24 31041051065 61024
25310 5561071081025
26456585 10513514526
27410610951310 19- 201027527
28 410751110215 28528554528
w=89 1011121314


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