Existence of $q$-Analogs of Steiner Systems

• DOI: 10.48550/arxiv.1304.1462
• Publication Date: 2013-01-01
• Authors: Michael Braun; Tuvi Etzion; Patric R. J. Öste...; Alexander Vardy; Alfred Wassermann
• Keywords: Steiner system; Centralizer and normalizer

Let $\F_q^n$ be a vector space of dimension $n$ over the finite field $\F_q$. A $q$-analog Steiner system (briefly, $q$-Steiner system), denoted $S_q[t,k,n]$, is set $S$ $k$-dimensional subspaces such that each $t$-dimensional subspace contained in exactly one element $S$. Presently, systems are known only for $t=1$, and trivial cases $t = k$ $k n$. Invthis paper, first nontrivial with >= 2$ constructed. Specifically, several nonisomorphic $S_2[2,3,13]$ found by requiring their automorphism groups contain normalizer Singer subgroup $\GL(13,2)$. This approach leads to an instance exact cover problem, which turns out have many solutions.