Semidefinite programming bounds for constant weight codes

Abstract

For nonnegative integers $n,d,w$, let $A(n,d,w)$ be the maximum size of a code $C\subseteq {\mathbb{F}}_2^n$ with 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 $A(22,8,w)$.

15 pages

References in corpus (3)

• Semidefinite bounds for nonbinary codes based on quadruples
• Block diagonalization for algebra's associated with block codes
• Semidefinite bounds for mixed binary/ternary codes

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• Uniqueness of codes using semidefinite programming