Subset sums and block designs in a finite vector space

Abstract

In this paper we settle the question of whether a finite-dimensional vector space {{mathcal {V}}} over {mathbb {F}}_p, with p an odd prime, and the family of all the k-sets of elements of {mathcal {V}} summing up to a given element x, form a 1-(v,k,lambda _1) or a 2-(v,k,lambda _2) block design, and, in either case, we find a closed form for lambda _i and characterize the automorphism group. The question is discussed also in the case where the elements of the k-sets are required to be all nonzero, as the two cases happen to be intrinsically inseparable. The “twin case” p=2, which has strict connections with coding theory, was completely discussed in a recent paper by G. Falcone and the present author.