Necessary conditions for the existence of 3-designs over finite fields with nontrivial automorphism groups

A q-design with parameters t-(v,k,lambda_t)_q is a pair (V, B) of the v-dimensional vector space V over GF(q) and a collection B of k-dimensional subspaces of V, such that each t-dimensional subspace of V is contained in precisely lambda_t members of B. In this paper we give new general necessary conditions on the existence of designs over finite fields with parameters 3-(v, k , lambda_3)_q with a prescribed automorphism group. These necessary conditions are based on a tactical decomposition of such a design over a finite field and are given in the form of equations for the coefficients of tactical decomposition matrices. In particular, they represent necessary conditions on the existence of q-analogues of Steiner systems admitting a prescribed automorphism group.