EXISTENCE OF q-ANALOGS OF STEINER SYSTEMS. (30th August 2016) - BRAUN, MICHAEL; ETZION, TUVI; ÖSTERGÅRD, PATRIC R. J.; VARDY, ALEXANDER; WASSERMANN, ALFRED
Abstract : Let F_q^n be a vector space of dimension n over the finite field F_q . A q -analog of a Steiner system (also known as a q - Steiner system ), denoted S_q(t, \!k, \!n), is a set S of k -dimensional subspaces of F_q^n such that each t -dimensional subspace of F_q^n 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 S_2(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.