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

2026-05-05

AI Summary

This study addresses the lack of tight bounds on the covering radius of binary even-strength orthogonal arrays. By integrating linear programming with constructive methods, the authors establish new upper bounds in two distinct cases. Furthermore, leveraging algebraic curve theory over finite fields and the dual structures of BCH, Melas, and Zetterberg codes, they derive improved lower bounds for three infinite families of orthogonal arrays. The proposed linear programming bound, which depends on the array’s cardinality and the location of farthest points, surpasses the classical Tietäväinen bound—particularly in non-tight settings. These results yield highly accurate estimates of the covering radius across multiple parameter regimes.

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{ä}v{ä}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.

Problem

• covering radius
• orthogonal arrays
• even strength
• binary codes
• linear programming bounds

Innovation

• orthogonal arrays
• covering radius
• linear programming bounds
• algebraic curves
• binary codes