Difference systems of sets and a collection of 3-subsets in a finite field of order p

A difference system of set (DSS) is a collection of t disjoint τi-subsets Qi, 0≤i≤t−1, of Zn such that every non-identity element of Zn appears at least ρ times in the multiset {a−b|a∈Qi,b∈Qj,0≤i,j≤t−1,i≠j}. A DSS is regular if τi is constant for 0≤i≤t−1, and a DSS is perfect if every element of Zn is contained exactly ρ times in the above multiset. In this paper, we consider a collection of 3-subsets of a finite field of a prime order p=ef+1 to be a DSS. We present a condition for which the collection forms a regular DSS and give a lower bound on the parameter ρ using cyclotomic numbers for e=3,4 and 6. For the same values of e, we also show a condition for which a collection of 3-subsets is a perfect DSS.