Asymptotic size of covering arrays: an application of entropy compression

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.