q-Steiner Systems and their Automorphisms

Abstract

The q-analog of t-designs and Steiner systems arises canonically from replacing sets of conventional t-designs by vector spaces over GF(q) and their orders with the dimensions. Thomas introduced these generalizations [21] and a few q-analogs of t-designs are known today. Minimal progress was made in constructing a q-Steiner system. In 2013, the rst nontrivial q-Steiner system was constructed S2[2; 3; 13] by Etzion [4] using certain automorphisms groups. This paper focuses on properties of 2-Steiner systems, in a general sense. The notion of an embedded 'skew' design is introduced and the consequences on existence are discussed. The smallest nontrivial S2[2; 3; n] that can exist is n = 7, and currently, its existence is unknown. Parameters from S2[2; 3; n] were applied to S2[2; 3; 7]. Curious observations of the relationship between points in a hyperplane and 5-spaces were made leading to the notion of a 'special point'. Automorphisms of 2-Steiner systems, S2[2; 3; n] and n = 7, of odd order are investigated and theoretic proofs of nonexis- tence is given.