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) = 2^n-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

• Mar. 2, 2001: The web version of the table in [2] was published.
• June 1, 2001: 6 updates by Mounits et al., from a preliminary version of [7].
• Oct. 28, 2003: 10 further updates by Mounits et al., from the published version of [7].
• July 8, 2004: 11 updates by Schrijver, from a preliminary version of [8].
• Jan. 5, 2015: 2 of the 2004 updates were corrected based on the published version of [8].
• Jan. 5, 2015: 1 update by Östergård, from [11].
• Jan. 5, 2015: 18 updates by Gijswijt et al., from [12].

Legend of superscripts

• 1-4: Theorem numbers in [2].
• S: Specific bounds described in the last paragraph of [2].
• M: From [7].
• T: From [8], Table I.
• O: From [11].
• G: From [12], Table I.

Bounds on A(n,d)