Abstract: | Let n and k be positive integers. Let Cq 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 (aij) with entries aij (0 ≤ i ≤ k−1, 0 ≤ j ≤ n−1) from Cq such that, for any two rows t and h (0 ≤ t < h ≤ k−1), every element of Cq occurs in the difference list
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. |