Semidefinite lower bounds for covering codes

2025-04-02

AI Summary

This paper addresses the problem of establishing lower bounds on the minimal size $ K_q(n,r) $ of a $ q $-ary covering code—i.e., the smallest number of length-$ n $ codewords over an alphabet of size $ q $ whose Hamming balls of radius $ r $ collectively cover the entire $ q $-ary space—with particular focus on the classical football pool problem $ K_3(n,1) $. Methodologically, we introduce the Lasserre hierarchy of semidefinite programming (SDP) to construct lower bounds, integrating symmetry reduction via group representation theory and novel SDP constraints. This approach substantially strengthens the classical matrix-cutting bound framework. Our contributions include new record lower bounds for $ K_q(n,r) $ across multiple parameter sets $ (q,n,r) $, most notably a significant improvement over the best previously known lower bound for $ K_3(n,1) $. These results advance both covering code theory and combinatorial optimization by providing a principled, computationally tractable method and new benchmark values.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 $r$.

Problem

• Develop semidefinite programming bounds for q-ary covering codes
• Improve lower bounds on K_q(n,r) using new constraints
• Enhance football pool problem solutions via symmetry reduction

Innovation

• Semidefinite programming for code bounds
• New constraints from Lasserre's hierarchy
• Symmetry reduction for sharper bounds