was defined as

$$\label{eq:resKiLa}
\{ B ~:~ B \in {{\mathbb{S}}},~ B \subset V \}~.$$

By these definitions, the derived design and residual design of a $q$-Steiner system are both designs over ${{\mathbb{F}}}_q$. This is on the positive side. On the negative side, the size of their union is significantly smaller than the size of the design from which they were derived.

We present now a different definition for the $q$-analog of a derived design and a residual design which solves this problem in the definition of [@KiLa15]. Let $u$ be the unit vector with the unique *one* in the last coordinate, and $V {\mbox{$\stackrel{\rm def}{=}$}}\{ (x,0) ~:~ x \in {{\mathbb{F}}}_q^{n-1} \}$. Also, for a subspace $B \subset {{\mathbb{F}}}_q^n$, let ${{\cal Z}}(B)$ be the subspace obtained from $B$, by removing the last coordinate of all the vectors in $B$. The derived and residual designs are defined by

$$\label{eq:derived}
\text{der} ({{\mathbb{S}}}) {\mbox{$\stackrel{\rm def}{=}$}}\{ {{\cal Z}}(B \cap V) ~:~ B \in {{\mathbb{S}}}, ~ u \in B \}~.$$

$$\label{eq:residual}
\text{res} ({{\mathbb{S}}}) {\mbox{$\stackrel{\rm def}{=}$}}\{ {{\cal Z}}(B) ~:~ B \in {{\mathbb{S}}}, ~ u \notin B \}~.$$

The two definitions of the derived design are equivalent, but there is a significant difference in the two definitions of 