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

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