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EXISTENCE OF $q$-ANALOGS OF STEINER SYSTEMS

Published online by Cambridge University Press:  30 August 2016

MICHAEL BRAUN
Affiliation:
Darmstadt University of Applied Sciences, Darmstadt, Germany; michael.braun@h-da.de
TUVI ETZION
Affiliation:
Technion, Haifa, Israel; etzion@cs.technion.ac.il
PATRIC R. J. ÖSTERGÅRD
Affiliation:
Aalto University, Aalto, Finland; patric.ostergard@aalto.fi
ALEXANDER VARDY
Affiliation:
University of California San Diego, La Jolla, California, USA Nanyang Technological University, Singapore; avardy@ucsd.edu
ALFRED WASSERMANN
Affiliation:
University of Bayreuth, Bayreuth, Germany; Alfred.Wassermann@uni-bayreuth.de

Abstract

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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.

MSC classification

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
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.
Copyright
© The Author(s) 2016

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EXISTENCE OF $q$-ANALOGS OF STEINER SYSTEMS
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