Asymptotic size of covering arrays: an application of entropy compression

TL;DR

This paper establishes upper bounds on the asymptotic size of covering arrays using entropy compression techniques, advancing understanding of their minimal size in relation to parameters t, v, and k.

Contribution

It introduces two new upper bounds on the asymptotic growth of covering array sizes employing an entropy compression approach based on the Lovász local lemma.

Findings

Derived two upper bounds on the asymptotic size of covering arrays.

Applied entropy compression to improve bounds on $CAN(t,k,v)$.

Enhanced theoretical understanding of covering array size growth.

Abstract

A covering array $CA(N; t,k,v)$ is an $N \times k$ array $A$ whose each cell takes a value for a $v$-set $V$ called an alphabet. Moreover, the set $V^t$ is contained in the set of rows of every $N \times t$ subarray of $A$. The parameter $N$ is called the size of an array and $CAN(t,k,v)$ denotes the smallest $N$ for which a $CA(N; t,k,v)$ exists. It is well known that $CAN(t,k,v) = {\rm \Theta}(\log_2 k)$~\cite{godbole_bounds_1996}. In this paper we derive two upper bounds on $d(t,v)=\limsup_{k \rightarrow \infty} \frac{CAN(t,k,v)}{\log_2 k}$ using the algorithmic approach to the Lov\'{a}sz local lemma also known as entropy compression.