Semidefinite programming bounds for Lee codes

TL;DR

This paper develops semidefinite programming bounds for Lee codes, providing new upper bounds on their maximum size and related graph parameters, with computational efficiency achieved through symmetry reductions.

Contribution

It introduces a semidefinite programming approach based on triples of codewords to bound Lee codes and related graph capacities, improving bounds for small parameters.

Findings

New upper bounds on $A_q^L(n,d)$ for various parameters.

Bounds on the independent set number of strong product powers of circular graphs.

The bounds are computationally efficient due to symmetry reductions.

Abstract

For $q,n,d \in \mathbb{N}$, let $A_q^L(n,d)$ denote the maximum cardinality of a code $C \subseteq \mathbb{Z}_q^n$ with minimum Lee distance at least $d$, where $\mathbb{Z}_q$ denotes the cyclic group of order $q$. We consider a semidefinite programming bound based on triples of codewords, which bound can be computed efficiently using symmetry reductions, resulting in several new upper bounds on $A_q^L(n,d)$. The technique also yields an upper bound on the independent set number of the $n$-th strong product power of the circular graph $C_{d,q}$, which number is related to the Shannon capacity of $C_{d,q}$. Here $C_{d,q}$ is the graph with vertex set $\mathbb{Z}_q$, in which two vertices are adjacent if and only if their distance (mod $q$) is strictly less than $d$. The new bound does not seem to improve significantly over the bound obtained from Lov\'asz theta-function, except for very small $n$.