${{\mathbb{F}}}_q^4$ to subspaces in ${{\mathbb{F}}}_q^6$. The construction of the residual $q$-Fano plane is presented in Section \[sec:construction\], where its correctness is also proved. Conclusions and future research are discussed in Section \[sec:conclude\]. In particular we indicate on the points in the construction in which there is flexibility to construct many different residual $q$-Fano planes.

Derived and Residual Designs {#sec:residual}
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For a design $S$ on a set $Q$, and an element $x \in Q$, the derived design is defined by $$\{ B \setminus \{ x \} ~:~ B \in S,~ x \in B \}~,$$ and the residual design is defined by $$\{ B ~:~ B \in S,~ x \notin B \}~.$$

In [@KiLa15] there is a simple definition for a $q$-analog of the derived design and the residual design. For this definition we choose an element $u \in {{\mathbb{F}}}_q^n$ and an $(n-1)$-subspace $V \subset {{\mathbb{F}}}_q^n$ such that ${{\left\langle { \{ u \} \cup V } \right\rangle}} = {{\mathbb{F}}}_q^n$, where ${{\left\langle {X} \right\rangle}}$ denote the linear span of $X$. The derived design of a design ${{\mathbb{S}}}$ over ${{\mathbb{F}}}_q$, was defined as

$$\label{eq:derKiLa}
\{ B \cap V ~:~ B \in {{\mathbb{S}}}, ~ u \in B \}~,$$

and the residual design of ${{\mathbb{S}}}$, 