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.

Let Aq(n, d, k) denote the maximum cardinality of a set C of k-dimensional subspaces of an n-dimensional vector space over the finite field of order q, F q, such that any two different subspaces U, W. C have a distance d(U, W) := dim(U + W) -dim(U n W) of at least d. Lower bounds on Aq(n, d, k) can be obtained by explicitly constructing corresponding sets C. When searching for such sets with a prescribed group of automorphisms, the search problem leads to instances of the maximum weight clique problem. The main focus is here on subgroups with small index in the normalizer of a Singer subgroup of GL(n, q). With a stochastic maximum weight clique algorithm and a systematic consideration of groups of the above mentioned type, new lower bounds on A2(8, 4, 4) and A2(n, 4, 3) for 8 = n = 11 are obtained.