New bounds on the covering radius of orthogonal arrays of even strength

TL;DR

This paper introduces new linear programming and constructive bounds for the covering radius of binary orthogonal arrays of even strength, improving upon existing bounds and analyzing specific families related to classical codes.

Contribution

The paper develops novel LP bounds for the covering radius of orthogonal arrays, considering different geometric scenarios, and applies algebraic curve techniques to specific array families.

Findings

New LP bounds outperform previous bounds for non-tight arrays.

Bounds are tighter for arrays related to BCH, Melas, and Zetterberg codes.

Close estimates of covering radius are achieved for certain array families.

Abstract

We obtain new linear programming (LP) and constructive bounds for the covering radius of binary orthogonal arrays of strength $2k$. Our LP bounds develop in two alternative scenarios. First, if a point $y \in F_2^n$, where the covering radius of some orthogonal array $C \subset F_2^n$ of strength $2k$ is realized, is such that the farthest point of $C$ to $y$ is not antipodal to $y$ we obtain a bound which is better than the Tiet{\"a}v{\"a}inen (or Fazekas-Levenshtein) bound for non-tight arrays (i.e., the cardinality strictly exceeds the Rao lower bound). Second, if all points where the covering radius is realized are such that their antipodes are in $C$, we obtain a bound which depends on the cardinality of $C$ and is again better whenever the orthogonal array is not tight. We further describe three infinite families of binary orthogonal arrays related to the duals of BCH, Melas, and Zetterberg codes. For these families, we derive lower bounds on the covering radius by applying techniques from algebraic curves over finite fields, while the improved linear programming methods developed in this paper provide upper bounds, leading in some cases to fairly close estimates.