EXISTENCE OF $q$-ANALOGS OF STEINER SYSTEMS. (30th August 2016) - BRAUN, MICHAEL; ETZION, TUVI; ÖSTERGÅRD, PATRIC R. J.; VARDY, ALEXANDER; WASSERMANN, ALFRED
Abstract : Let $\mathbb{F}_{q}^{n}$ be a vector space of dimension $n$ over the finite field $\mathbb{F}_{q}$ . A  $q$ -analog of a Steiner system (also known as a $q$ - Steiner system ), denoted ${\mathcal{S}}_{q}(t, \!k, \!n)$, is a set ${\mathcal{S}}$ of $k$ -dimensional subspaces of $\mathbb{F}_{q}^{n}$ such that each $t$ -dimensional subspace of $\mathbb{F}_{q}^{n}$ is contained in exactly one element of ${\mathcal{S}}$ . 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 $t\, \geqslant \, 2$ are constructed. Specifically, several nonisomorphic $q$ -Steiner systems ${\mathcal{S}}_{2}(2, 3, 13)$ are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of $\text{GL}(13, 2)$ . This approach leads to an instance of the exact cover problem, which turns out to have many solutions.