Bounds for unrestricted binary codes

This table gives, for all parameters n and d, an interval for the maximum size A(n,d) of an unrestricted binary block code. Unrestricted means in this context that no further constraint (such as linearity, constant weight, etc.) is imposed on the binary codes. A single value indicates that the lower and upper bounds coincide: A(n,d) is known exactly in such cases. Superscripts denote the methods by which the upper bounds were obtained, as detailed in the legend below.

The table gives the best known bounds on A(n,d) for all n up to 28 and all even d <= n. For odd values of d, A(n,d) = A(n+1,d+1). The table can be extended to larger and smaller values of d by noting that A(n,d) = 2 for 2n/3 < d <= n and A(n,2) = 2n-1 for all n.

Tables for lower bounds of binary codes are available on several websites. The lower bounds in the table below were copied from [LRS] in January 2001 and are not regularly updated. See [B] for updated lower bounds and details on how they were obtained.

Table history

Please report further updates or corrections.

Legend of superscripts

Bounds on A(n,d)

d=4681012141618
n=4 21n=4
521 5
641216
7 81217
8161 21218
9204 41219
10 40161212110
1172S12121 2111
12144S 24141212112
13256332S 41212113
1451236438121 212114
15 10242128316141 212115
16 20482256232141 21212116
172720 - 32763256 - 340S36O612121 2117
185312 - 65521 512 - 673G64 - 72S101 4121212118
1910496 - 1310411024 - 1237G128 - 135G2014121 212119
2020480 - 262081 2048 - 2279G256G401 6121212120
2136864 - 43688M2560 - 4096S 512S42 - 47G8141 212121
2273728 - 87376M 4096 - 694141024350 - 84G 12141212122
23147456 - 173015M8192 - 13674G 2048376 - 150424141 212123
24294912 - 344308216384 - 24106440962128 - 268G4816141 2124
25524288 - 599184M16384 - 47998T4096 - 5421G176 - 466G52 - 55G81412125
261048576 - 1198368M32768 - 84260M4096 - 9275G270 - 836G64 - 96G 141412126
272097152 - 2396736M65536 - 157285M8192 - 17099G 512 - 1585G128 - 1694281 614127
284194304 - 4793472M131072 - 291269416384 - 32151T 1024 - 2817G178 - 2883561 814128
d=4681012 141618


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