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 $S(D) := \{C \in \mathcal{C}_k \mid D \subseteq C, |D| +2|C\setminus D| \leq k\}$. 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 $\mathcal{C}_k \rightarrow \mathbb{R}$ such that $x(\emptyset) = 1$ and $x(C) = 0$ if $C$ has minimum distance less than $d$, and such that the $S(D)\times S(D)$ matrix $(x(C\cup C'))_{C,C' \in S(D)}$ 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