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

• Aug. 10, 2000: The web version of the tables in [1] was published.
• July 8, 2004: 33 updates by Schrijver, from a preliminary version of [8].
• Jan. 5, 2015: 47 further updates by Schrijver, plus a few corrections, from the published version of [8].
• Jan. 5, 2015: 35 updates by Östergård, from [10].

Legend of superscripts

• 5-21: Theorem numbers in [1].
• T: From [8], Table II.
• O: From [10], Tables IX–XI and Theorem 3.

Bounds on A(n,4,w)

Bounds on A(n,6,w)

Bounds on A(n,8,w)

Bounds on A(n,10,w)

Bounds on A(n,12,w)

Bounds on A(n,14,w)