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EXISTENCE OF $q$ -ANALOGS OF STEINER SYSTEMS

  • MICHAEL BRAUN (a1), TUVI ETZION (a2), PATRIC R. J. ÖSTERGÅRD (a3), ALEXANDER VARDY (a4) (a5) and ALFRED WASSERMANN (a6)...
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.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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