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$q$-ANALOGS OF STEINER SYSTEMS
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.
