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 V over F_p, with p an odd prime, and the family of all the k-sets of elements of V summing up to a given element x, form a 1-(v, k, λ_1) or a 2-(v, k, λ_2) block design, and, in either case, we find a closed form for λ_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.