in other areas, such as network coding [@EtSt16]. The appropriate code in random network coding is a [collection of subspaces]{} of ${{\mathbb{F}}}_q^n$ that are well-separated according to a metric defined on the Grassmannian. Consequently, a $q$-Steiner system ${{\mathbb{S}}}_q(t,k,n)$ can be thought of as an *optimal code* for error-correction in networks. For more details on this, see [@EtVa11; @KoKs08].

Following the work of Cameron [@Cam74; @Cam74a] and Delsarte [@Del76], the first of nontrivial $t$-designs over ${{\mathbb{F}}}_q$ were found by Thomas [@Tho87] in 1987. Today,  to the efforts of many authors [@BKL; @FLV14; @KiLa15; @MMY95; @RaSi89; @Suz90; @Suz90a; @Suz92; @Tho96], numerous such examples are known.

However, the situation is very different for $q$-Steiner systems. They are known to exist in the trivial cases $t = k$ or $k=n$, and in the case where $t = 1$ and $k$ divides $n$. In the latter case, $q$-Steiner systems coincide with the classical notion of *spreads* in projective geometry [@vLWi92 Chapter24]. Some 40 years ago, Beutelspacher [@Beu78] asked whether nontrivial $q$-Steiner systems with $t \geq 2$ exist, and this question has tantalized mathematicians ever since. The problem has been studied by numerous authors [@AAK01; @EtVa11a; @Met99; @ScEt02; @Tho87; @Tho96], without much progress toward constructing such 