Constructions of difference covering arrays

Abstract

A difference covering array with parameters k, n and q, or a DCA( k, n; q) for short, over a group ( G,•) of order q is defined to be a k× n array ( a ij ) with entries a ij (0⩽i⩽k−1,0⩽ j⩽n−1) from G such that, for any two distinct rows t and h (0⩽ t< h⩽ k−1), every element of G occurs in the difference list {d hj•d tj −1 : j=0,1,…,n−1} at least once. It is clear that n⩾ q in a DCA( k, n; q). The equality holds if and only if a ( q, k,1) difference matrix exists. It is well known that a ( q, k,1) difference matrix does not exist, whenever q≡2 ( mod 4) and k⩾3. Thus, we have n⩾ q+1 for these values of k and q. In this article, several constructive techniques for DCAs are presented, and used to solve completely the existence problem for a DCA(4, q+1; q) with q≡2 ( mod 4) . This complements the study for difference matrices in literature. The result is also useful in encoding systematic authentication codes, as well as in software testing and data compression problems.