$t$-Covering Arrays Generated by a Tiling Probability Model

TL;DR

This paper introduces a new probabilistic model using tiling and the Lovász Local Lemma to improve bounds on the minimal size of $t$-covering arrays, which are combinatorial structures ensuring coverage of all $t$-tuples.

Contribution

It presents a novel tiling-based probability model combined with the Lovász Local Lemma to derive tighter upper bounds on the size of $t$-covering arrays.

Findings

Improved upper bounds on the smallest number of columns $N$ for $t$-covering arrays.

Development of a new tiling probability model for combinatorial array analysis.

Application of the Lovász Local Lemma to combinatorial array construction.

Abstract

A $t-\a$ covering array is an $m\times n$ matrix, with entries from an alphabet of size $\alpha$, such that for any choice of $t$ rows, and any ordered string of $t$ letters of the alphabet, there exists a column such that the "values" of the rows in that column match those of the string of letters. We use the Lov\'asz Local Lemma in conjunction with a new tiling-based probability model to improve the upper bound on the smallest number of columns $N=N(m,t,\alpha)$ of a $t-\a$ covering array.