Determining Covering Array Numbers via Balanced Covering Arrays

TL;DR

This paper determines five previously unknown covering array numbers by leveraging properties of balanced covering arrays and computational methods, advancing the understanding of optimal combinatorial designs.

Contribution

It introduces a novel approach using balanced covering arrays to compute previously unknown covering array numbers, filling longstanding gaps in combinatorial design theory.

Findings

Determined five new covering array numbers (CANs).

Generalized non-existence results from balanced to covering arrays.

Provided computational bounds for these CANs.

Abstract

In this article we determine five previously unknown covering array numbers (CANs). We do so using properties of so called balanced covering arrays together with a computational result for these. The balance properties allow us to generalize the (computational) non-existence result for balanced covering arrays to covering arrays. Covering arrays are combinatorial designs that can be considered generalizations of orthogonal arrays, when dropping the restriction that the considered $t$-tuples appear exactly $\lambda$ times, and instead require them to appear at least $\lambda$ times. While this generalization renders the existence of covering arrays trivial, it raises the question for their optimality, respectively the smallest number of rows, the CAN, for which a certain covering array exists. The CANs determined in this paper were tightly bound for decades, but remained ultimately unknown.