Structures and lower bounds for binary covering arrays

TL;DR

This paper investigates the structure and lower bounds of binary covering arrays, extending known results for specific parameters and establishing uniqueness and construction properties for certain cases.

Contribution

It provides new bounds for binary covering arrays with higher strength and demonstrates that some arrays are uniquely determined, advancing the theoretical understanding of their structure.

Findings

Improved lower bounds for binary 3-covering arrays.

Identification of conditions under which arrays are uniquely determined.

Connection between maximal and general binary 2-covering arrays.

Abstract

A $q$-ary $t$-covering array is an $m \times n$ matrix with entries from $\{0, 1, ..., q-1\}$ with the property that for any $t$ column positions, all $q^t$ possible vectors of length $t$ occur at least once. One wishes to minimize $m$ for given $t$ and $n$, or maximize $n$ for given $t$ and $m$. For $t = 2$ and $q = 2$, it is completely solved by R\'enyi, Katona, and Kleitman and Spencer. They also show that maximal binary 2-covering arrays are uniquely determined. Roux found the lower bound of $m$ for a general $t, n$, and $q$. In this article, we show that $m \times n$ binary 2-covering arrays under some constraints on $m$ and $n$ come from the maximal covering arrays. We also improve the lower bound of Roux for $t = 3$ and $q = 2$, and show that some binary 3 or 4-covering arrays are uniquely determined.