Determining Covering Array Numbers via Balanced Covering Arrays

TL;DR

The paper tackles the problem of determining minimal row counts for covering arrays by leveraging balanced covering arrays and computational classification. It defines and uses $CA$ and $CAN$ formalisms, introducing $(\boldsymbol{\lambda},\mathbf{y})$-balanced CAs to transfer non-existence results to ordinary CAs. The main contributions are new CAN lower bounds $CAN(3,k,2) \ge 18$ for $k \in \{17,18,19,20\}$ and $CAN(4,18,2)=36$, along with the optimality of five known CAs, including an explicit $CA(18;3,20,2)$. Additionally, a non-existence result for $CA(17;3,17,2)$ is established via balance-vector inequalities and exhaustive computation. Overall, the work tightens the landscape of CAN values and demonstrates a robust method combining balance theory with computation to identify optimal covering arrays and guide future CAN determinations.

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 $λ$ times, and instead require them to appear at least $λ$ 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.

Figures (1)

• Figure 1: An optimal $\mathsf{CA}(18;3,20,2)$.

Theorems & Definitions (4)

• Definition 1: Balanced CA, balCApaper
• Theorem 1
• Corollary 1
• Corollary 2