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.

中文翻译：

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施泰纳系统的类比的存在

让$\mathbb{F}_{q}^{n}$是维向量空间$n$在有限域上$\mathbb{F}_{q}$. 一种$q$- Steiner 系统的模拟（也称为$q$-施泰纳系统)，表示${\mathcal{S}}_{q}(t,\!k,\!n)$, 是一个集合${\mathcal{S}}$的$k$维子空间$\mathbb{F}_{q}^{n}$使得每个$t$-维子空间$\mathbb{F}_{q}^{n}$恰好包含在一个元素中${\mathcal{S}}$. 目前，$q$-Steiner 系统仅适用于$t\,=\,1\!$，并且在琐碎的情况下$t\,=\,k$和$k\,=\,n$. 在本文中，第一个非平凡的$q$-施泰纳系统$t\,\geqslant\,2$被构造。具体来说，几个非同构$q$-施泰纳系统${\mathcal{S}}_{2}(2,3,13)$通过要求它们的自同构群包含 Singer 子群的归一化器来找到$\文本{GL}(13,2)$. 这种方法导致了一个精确覆盖问题的实例，结果证明它有很多解决方案。