Semidefinite bounds for nonbinary codes based on quadruples

Abstract

For nonnegative integers $q,n,d$, let $A_q(n,d)$ denote the maximum cardinality of a code of length $n$ over an alphabet $[q]$ with $q$ letters and with minimum distance at least $d$. We consider the following upper bound on $A_q(n,d)$. For any $k$, let \CC_k be the collection of codes of cardinality at most $k$. Then $A_q(n,d)$ is at most the maximum value of $\sum_{v\in[q]^n}x(\{v\})$, where $x$ is a function \CC_4\to R_+ such that $x(\emptyset)=1$ and $x(C)=0$ if $C$ has minimum distance less than $d$, and such that the \CC_2\times\CC_2 matrix (x(C\cup C'))_{C,C'\in\CC_2} is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in $n$. It yields the new upper bounds $A_4(6,3)\leq 176$, $A_4(7,4)\leq 155$, $A_5(7,4)\leq 489$, and $A_5(7,5)\leq 87$