A Note on the Existence of Non-simple Designs over Finite Fields

Designs over finite fields arise by replacing finite sets by vector spaces and orders of sets by dimensions of vector spaces. More formally, a t− (v, k, λ; q) design is a collection of k-subspaces over Fq , called blocks, such that each t-subspace is contained in exactly λ blocks. Such a design over a finite field is called simple, if no repeated blocks occur in the collection. Otherwise, the design is called non-simple. In this paper we prove that for every parameter set 0 < t < k < v− t and each subgroup G of the general linear group GL(v, q) a non-simple t− (v, k, λ; q) design exists for some appropriate λ > 0 admitting G as a group of automorphisms.