$n$) for which the existence question of $q$-Steiner systems is not settled is the parameters for the $q$-analog of the Fano plane, i.e. the $q$-Steiner systems ${{\mathbb{S}}}_q(2,3,7)$, which will be called also in this paper the $q$-*Fano plane*. There was a lot of effort to find whether the $q$-Fano plane, especially for $q=2$, exists or does not exist, e.g. [@BKN15; @DePh17; @EtVa11a; @Tho96]. All these attempts did not provide any answer to the existence question. It was proved recently in [@BKN15] that if such system exists for $q=2$, then its automorphism group has a small order. In [@Etz15] a different approach to consider $q$-Steiner systems was given. This approach is based on puncturing a possible existing $q$-Steiner systems and considering the parameters of the structure derived from the punctured systems. Properties of the $q$-Fano plane based on this approach were also discussed. This approach led to the results in the current paper.

In this paper we present a construction for a design with the same parameters as the design derived from a $q$-Fano plane, the residual $q$-Fano plane. The constructed design will be also called the residual $q$-Fano plane. The construction has many places in which there is flexibility for 