Semidefinite bounds for mixed binary/ternary codes

Abstract

For nonnegative integers $n_2,n_3$ and $d$, let $N(n_2,n_3,d)$ denote the maximum cardinality of a code of length $n_2+n_3$, with $n_2$ binary coordinates and $n_3$ ternary coordinates (in this order) and with minimum distance at least $d$. For a nonnegative integer $k$, let ${\mathcal{C}}_k$ denote the collection of codes of cardinality at most $k$. For $D\in {\mathcal{C}}_k$, define $S(D):=\{C\in {\mathcal{C}}_k\mid D\subseteq C,|D|+2|C\setminus D|\le k\}$. Then $N(n_2,n_3,d)$ is upper bounded by the maximum value of ${\sum }_{v\in [2{]}^{n_2}[3{]}^{n_3}}x(\{v\})$, where $x$ is a function ${\mathcal{C}}_k\to \mathbb{R}$ such that $x(\varnothing )=1$ and $x(C)=0$ if $C$ has minimum distance less than $d$, and such that the $S(D)\times S(D)$ matrix $(x(C\cup C^{\prime }){)}_{C,C^{\prime }\in S(D)}$ is positive semidefinite for each $D\in {\mathcal{C}}_k$. By exploiting symmetry, the semidefinite programming problem for the case $k=3$ is reduced using representation theory. It yields $135$ new upper bounds that are provided in tables

12 pages; some typos have been fixed. Accepted for publication in Discrete Mathematics

References in corpus (1)

• Semidefinite bounds for nonbinary codes based on quadruples

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