Semidefinite lower bounds for covering codes
Dion Gijswijt, Sven Polak
TL;DR
This work advances lower-bounding techniques for covering codes by strengthening SDP-based bounds on $K_q(n,r)$ through Lasserre-inspired semidefinite constraints and explicit symmetry reduction. It combines a matrix-cut SDP framework with a detailed block-diagonalization of the Terwilliger algebra for binary and nonbinary Hamming schemes to drastically reduce problem size while tightening the bound. The authors derive both binary and nonbinary symmetry-reduced SDPs and show substantial improvements across many $(q,n,r)$, including new records in several cases. Computational results are reported in comprehensive tables, with high-precision numerical solutions and public code availability, underscoring the practical impact for covering codes and related combinatorial bounds.
Abstract
Let $K_q(n,r)$ denote the minimum size of a $q$-ary covering code of word length $n$ and covering radius $r$. In other words, $K_q(n,r)$ is the minimum size of a set of $q$-ary codewords of length $n$ such that the Hamming balls of radius $r$ around the codewords cover the Hamming space $\{0,\ldots,q-1\}^n$. The special case $K_3(n,1)$ is often referred to as the football pool problem, as it is equivalent to finding a set of forecasts on $n$ football matches that is guaranteed to contain a forecast with at most one wrong outcome. In this paper, we build and expand upon the work of Gijswijt (2005), who introduced a semidefinite programming lower bound on $K_q(n,r)$ via matrix cuts. We develop techniques that strengthen this bound, by introducing new semidefinite constraints inspired by Lasserre's hierarchy for 0-1 programs and symmetry reduction methods, and a more powerful objective function. The techniques lead to sharper lower bounds, setting new records across a broad range of values of $q$, $n$, and $r$.