Perfect difference systems of sets and the unit group of ZpnZpn

A perfect (v,{|Di|:0≤i≤s−1},ρ)(v,{|Di|:0≤i≤s−1},ρ) difference system of sets is a collection of s   disjoint subsets Di,0≤i≤s−1Di,0≤i≤s−1 of Zv, such that every non-zero element of Zv appears exactly ρρ times in the multiset {a−b:a∈Di,b∈Dj,0≤i≠j≤s−1}{a−b:a∈Di,b∈Dj,0≤i≠j≤s−1}. In this paper, we give two recursive constructions of perfect DSS   s by using the unit group of ZpnZpn, and obtain some infinite classes of perfect DSS   s over ZpnZpn.