EXISTENCE OF $q$ -ANALOGS OF STEINER SYSTEMS
Abstrak
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.
Topik & Kata Kunci
Penulis (5)
Michael Braun
T. Etzion
P. Östergård
A. Vardy
A. Wassermann
Akses Cepat
- Tahun Terbit
- 2013
- Bahasa
- en
- Total Sitasi
- 148×
- Sumber Database
- Semantic Scholar
- DOI
- 10.1017/fmp.2016.5
- Akses
- Open Access ✓