Cyclic Difference Packing and Covering Arrays

Abstract

Let n and k be positive integers. Let C _q be a cyclic group of order q. A cyclic difference packing (covering) array, or a CDPA(k, n; q) (CDCA(k, n; q)), is a k × n array (a _ij ) with entries a _ij (0 ≤ i ≤ k−1, 0 ≤ j ≤ n−1) from C _q such that, for any two rows t and h (0 ≤ t < h ≤ k−1), every element of C _q occurs in the difference list ${\Delta}_{th} = {d_{hj}- d_{tj}: j = 0, 1, \dots, n-1}$ at most (at least) once. When q is even, then n ≤ q−1 if a CDPA(k, n; q) with k ≥ 3 exists, and n ≥ q+1 if a CDCA(k, n; q) with k ≥ 3 exists. It is proved that a CDCA(4, q+1; q) exists for any even positive integers, and so does a CDPA(4, q−1; q) or a CDPA(4, q−2; q). The result is established, for the most part, by means of a result on cyclic difference matrices with one hole, which is of interest in its own right.