Perfect difference systems of sets and Jacobi sums

A perfect (v,{ki∣1≤i≤s},ρ)(v,{ki∣1≤i≤s},ρ) difference system of sets (DSS) is a collection of ss disjoint kiki-subsets DiDi, 1≤i≤s1≤i≤s, of any finite abelian group GG of order vv such that every non-identity element of GG appears exactly ρρ times in the multiset {a−b∣a∈Di,b∈Dj,1≤i≠j≤s}{a−b∣a∈Di,b∈Dj,1≤i≠j≤s}. In this paper, we give a necessary and sufficient condition in terms of Jacobi sums for a collection {Di∣1≤i≤s}{Di∣1≤i≤s} defined in a finite field FqFq of order q=ef+1q=ef+1 to be a perfect (q,{ki∣1≤i≤s},ρ)(q,{ki∣1≤i≤s},ρ)-DSS, where each DiDi is a union of cyclotomic cosets of index ee (and the zero 0∈Fq0∈Fq). Also, we give numerical results for the cases e=2,3e=2,3, and 4.