system. This property does not exist in the definition given in (\[eq:resKiLa\]). A third argument is a consequence of the next theorem.

The union of the derived $q$-Steiner system and the residual $q$-Steiner system was called in [@Etz15], the punctured (or 1-punctured) $q$-Steiner system. But, no such system was constructed in [@Etz15]. In the exposition given in [@Etz15] it was proved that

\[thm:DerRes\] If ${{\mathbb{S}}}$ is a $q$-Steiner system ${{\mathbb{S}}}_q(t,k,n)$, then the derived system contains exactly $\frac{{\nsbinom{{n-1}}{{t-1}}_{q}}}{{\nsbinom{{k-1}}{{t-1}}_{q}}}$ distinct $(k-1)$-subspaces which form a $q$-Steiner system ${{\mathbb{S}}}_q(t-1,k-1,n-1)$. Each $t$-subspace of ${{\mathbb{F}}}_q^{n-1}$ which is contained in a $(k-1)$-subspace of $\text{der} ({{\mathbb{S}}})$ is not contained in any of the $k$-subspaces of $\text{res}({{\mathbb{S}}})$. Each $t$-subspace of ${{\mathbb{F}}}_q^{n-1}$ which is not contained in a $(k-1)$-subspace of $\text{der}({{\mathbb{S}}})$, appears exactly $q^t$ times in the $k$-subspaces of $\text{res}({{\mathbb{S}}})$.

We will now define any two sets of subspaces which satisfy the properties given in Theorem \[thm:DerRes\] as the derived design and the residual design for a $q$-Steiner system ${{\mathbb{S}}}_q(t,k,n)$ (but do not depend on the existence of a $q$-Steiner system ${{\mathbb{S}}}_q(t,k,n)$). For a $q$-Steiner system ${{\mathbb{S}}}_q (t,k,n)$ these definitions are given as follows:

-   A *derived $q$-Steiner system* for a $q$-Steiner system ${{\mathbb{S}}}_q (t,k,n)$ is a $q$-Steiner system ${{\mathbb{S}}}_q 