Constructions of difference covering arrays

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 (aij) with entries aij (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 {dhj•dtj−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.