$q$-Steiner systems. In particular, Thomas [@Tho96] showed in 1996 that certain kinds of ${{\mathbb{S}}}_2(2,3,7)$ $q$-Steiner systems (the smallest possible example) cannot exist. Three years later, Metsch [@Met99] conjectured that nontrivial $q$-Steiner systems with $t \geq 2$ do not exist in general. In contrast to this conjecture, a $q$-Steiner system ${{\mathbb{S}}}_2 (2,3,13)$ was constructed recently [@BEOVW]. In fact, once one such system was found, other nonisomorphic systems with the same parameters were found.

Similarly, to Steiner systems, simple necessary divisibility conditions for the existence of a given $q$-Steiner system were developed [@ScEt02; @Suz90].

\[thm:derived\] If a $q$-Steiner system ${{\mathbb{S}}}_q (t,k,n)$ exists, then for each $i$, $1 \leq i \leq t-1$, a $q$-Steiner system ${{\mathbb{S}}}_q(t-i,k-i,n-i)$ exists.

\[cor:ness\] If a $q$-Steiner system ${{\mathbb{S}}}_q(t,k,n)$ exists, then for all $0 \leq i \leq t-1$, $$\frac{{\nsbinom{{n-i}}{{t-i}}_{q}}}{{\nsbinom{{k-i}}{{t-i}}_{q}}}$$ must be integers.

Deriving new designs from designs in general and $q$-Steiner systems in particular is an important direction to find new designs and to exclude the possible existence of other designs. Using $q$-analog of the derived design and the residual designs it was proved that sometimes the necessary conditions for the existence of a $q$-Steiner system ${{\mathbb{S}}}_q(t,k,n)$ are not sufficient [@KiLa15]. The first set of parameters ($t$, $k$, and 