Improved Semidefinite Programming Bound on Sizes of Codes

Abstract

Let $A(n,d)$ (respectively $A(n,d,w)$) be the maximum possible number of codewords in a binary code (respectively binary constant-weight $w$ code) of length $n$ and minimum Hamming distance at least $d$. By adding new linear constraints to Schrijver's semidefinite programming bound, which is obtained from block-diagonalising the Terwilliger algebra of the Hamming cube, we obtain two new upper bounds on $A(n,d)$, namely $A(18,8)\le 71$ and $A(19,8)\le 131$. Twenty three new upper bounds on $A(n,d,w)$ for $n\le 28$ are also obtained by a similar way.