Uniqueness of codes using semidefinite programming

Abstract

For n, d, w∈ N, let A(n, d, w) denote the maximum size of a binary code of word length n, minimum distance d and constant weight w. Schrijver recently showed using semidefinite programming that A(23 , 8 , 11) = 1288 , and the second author that A(22 , 8 , 11) = 672 and A(22 , 8 , 10) = 616. Here we show uniqueness of the codes achieving these bounds. Let A(n, d) denote the maximum size of a binary code of word length n and minimum distance d. Gijswijt et al. showed that A(20 , 8) = 256. We show that there are several nonisomorphic codes achieving this bound, and classify all such codes with all distances divisible by 4.