On the automorphism group of a binary q-analog of the Fano plane

An Sq[t,k,v] q-Steiner system is a collection of k-dimensional subspaces of the v-dimensional vector space Fq over the finite field Fq with q elements, called blocks, such that each t-dimensional subspace of Fq is contained in exactly one block. The smallest admissible parameters for which a q-Steiner system could exist is S2[2,3,7]. Up to now the issue whether q-Steiner systems with these parameters exist or not is still unsolved. In this paper we investigate the automorphism group of a putative S2[2,3,7] q-Steiner system. We conclude that in case of existence the automorphism group is cyclic and of order at most 4. MSC2010—51E10, 51E20, 05B05. Keywords—Random network codes, designs over finite fields, q-Steiner systems, automorphism group.