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arXiv · 1304.1462

Existence of $q$-Analogs of Steiner Systems

Michael Braun·Tuvi Etzion·Patric Ostergard·Alexander Vardy·Alfred Wassermann
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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 (briefly, 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 $k = n$. Invthis 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.

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Michael Braun, Tuvi Etzion, Patric Ostergard, Alexander Vardy, Alfred Wassermann. 2013-05-07. Existence of $q$-Analogs of Steiner Systems. https://arxiv.org/abs/1304.1462

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