combinatorics of sets could be regarded as the limiting case $q \to 1$ of combinatorics of vector spaces over the finite field ${{\mathbb{F}}}_q$. Indeed, there is a strong analogy between subsets of a set and subspaces of a vector space, expounded by numerous authors—see [@Cohn; @GR; @Wang] and references therein. It is therefore natural to ask which combinatorial structures can be generalized from sets (the $q \to 1$ case) to vector spaces over ${{\mathbb{F}}}_q$. For $t$-designs and Steiner systems, this question was first studied by Cameron [@Cam74; @Cam74a] and Delsarte [@Del76] in the early 1970s. Specifically, let ${{\mathbb{F}}}_q^n$ be a vector space of dimension $n$ over the finite field ${{\mathbb{F}}}_q$. Then a *$t$-$(n,k,\lambda)$ design over ${{\mathbb{F}}}_q$* is defined in [@Cam74; @Cam74a; @Del76] as a collection of $k$-subspaces of ${{\mathbb{F}}}_q^n$, called *blocks*, such that each $t$-subspace of ${{\mathbb{F}}}_q^n$ is contained in exactly $\lambda$ blocks. Such $t$-designs over ${{\mathbb{F}}}_q$ are the $q$-analogs of conventional combinatorial designs. By analogy with the $q \to 1$ case, a $t$-$(n,k,1)$ design over ${{\mathbb{F}}}_q$ is said to be a *$q$-Steiner system*, and denoted ${{\mathbb{S}}}_q(t,k,n)$.

[**Remark.**]{} We observe that $q$-analogs of designs and Steiner systems are not only of interest in their own right, but also arise naturally 