Subset sums and block designs in a finite vector space

Abstract

AbstractIn this paper we settle the question of whether a finite-dimensional vector space $${{\mathcal {V}}}$$ V over $${\mathbb {F}}_p,$$ F p , with p an odd prime, and the family of all the k-sets of elements of $${\mathcal {V}}$$ V summing up to a given element x,  form a 1-$$(v,k,\lambda _1)$$ ( v , k , λ 1 ) or a 2-$$(v,k,\lambda _2)$$ ( v , k , λ 2 ) block design, and, in either case, we find a closed form for $$\lambda _i$$ 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” $$p=2,$$ p = 2 , which has strict connections with coding theory, was completely discussed in a recent paper by G. Falcone and the present author.