Semidefinite bounds for nonbinary codes based on quadruples

Bart Litjens; Sven Polak; Alexander Schrijver

doi:10.1007/s10623-016-0216-5

paper

Semidefinite bounds for nonbinary codes based on quadruples

arXiv:1602.02531 · doi:10.1007/s10623-016-0216-5

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 $A_{5} (7, 5) \leq 87$ .

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Block diagonalization for algebra's associated with block codes

Semidefinite bounds for nonbinary codes based on quadruples

Abstract

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