Semidefinite Programming Bounds for Constant-Weight Codes.

For nonnegative integers $n$ , $d$ , and $w$ , let $A(n,d,w)$ be the maximum size of a code $C \subseteq \mathbb {F}_{2}^{n}$ with a constant weight $w$ and minimum distance at least $d$. We consider two semidefinite programs based on quadruples of code words that yield several new upper bounds on $A(n,d,w)$. The new upper bounds imply that $A(22,8,10)=616$ and $A(22,8,11)=672$. Lower bounds on $A(22,8,10)$ and $A(22,8,11)$ are obtained from the $(n,d)=(22,7)$ shortened Golay code of size 2048. It can be concluded that the shortened Golay code is a union of constant-weight $w$ codes of sizes $A(22,8,w)$. [ABSTRACT FROM AUTHOR]