On (g,4;1)-difference matrices

Let G be an abelian group of order g. A difference matrix based on G, denoted (g, k; 1)-DM, is a k x g matrix A = [a(ij)], aij in G, such that for each 1 <= r < s <= k, the differences a(rj) - a(sj), 1 <= j <= g, comprise all the elements of G. If G = Z(g), the difference matrix is called cyclic and denoted by (g, k; 1)-CDM. Motivated by the construction of g-fan H(4, g, 4, 3), we consider the existence of (g, 4; 1)-DMs. It is proved that a (g, 4; 1)-DM exists if and only if g >= 4 and g not equivalent to 2(mod 4). Some new results on (g, k; 1)-CDMs are also obtained, which are useful in the construction of both optical orthogonal codes and Z-cyclic whist tournaments. (c) 2005 Elsevier B.V. All rights reserved.