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.
Ultrafast adiabatic quantum algorithm for the NP-complete exact cover problem