Constructions Of (4m, 4, 4)-PCDPs

A Partitioned cyclic difference packing (PCDP), or a (v,Κ,λ)-PCDP inΖ_v, is a partition D={D₀,D₁,…,D_m－1} ofΖ_v into m subsets (called base blocks) such that the differences from these base blocks cover each nonzero residue (mod v) at mostλtimes. Let v,λ, m be positive integers, we denote byρ(v,m) the minimum index such that a (v,Κ,λ)-PCDP exists. And we call a (v,Κ,λ)-PCDP optimal if its indexλ=ρ(v,m). Such PCDPs can be used directly to produce optimal frequency hopping sequences and comma free codes. It is known that for any positive m and v = mμ＋(?) with 0≤(?)≤m－1,ρ(v,m)≥μ. Chee, Ling and Yin has studied the existence and the construction of the optimal PCDP attaining the lower bound of above inequality. Especially, a complete solution to the existence problem for the optimal (3m, [3^m],3)-PCDP. In this paper, we study the existence problem for the optimal (4m, [4^m],4)-PCDP attaining the lower bound of above inequality. With cyclic difference matrix and cyclic holey difference matrix we tackle the existence of a (4m, [4^m], 4)-PCDPs (m(?)0 (mod 4)). For (4m, [4^m], 4)-PCDPs (m≡0 (mod 4)), we present some constructive methods and existence results.