Semidefinite lower bounds for covering codes
Year: 2025
Type: preprint
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 $r$.
Source: arXiv (Cornell University)
Authors: Gijswijt, Dion, Polak, Sven
Institutions: Delft University of Technology, Tilburg University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6947849934
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