EXISTENCE OF q-ANALOGS OF STEINER SYSTEMS

Let Fn q be a vector space of dimension n over the finite field Fq . A q-analog of a Steiner system (also known as a q-Steiner system), denoted Sq(t,k,n), is a set S of k-dimensional subspaces of Fn q such that each t-dimensional subspace of Fn q is contained in exactly one element of S . Presently, q-Steiner systems are known only for t = 1, and in the trivial cases t = k and k= n. In this paper, the first nontrivial q-Steiner systems with t > 2 are constructed. Specifically, several nonisomorphic q-Steiner systems S2(2, 3, 13) are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of GL(13, 2). This approach leads to an instance of the exact cover problem, which turns out to have many solutions. 2010 Mathematics Subject Classification: 51E10 (primary); 05E20 (secondary)