Perfect difference systems of sets and Jacobi sums

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