many choices which lead to a construction of many such designs. Our definition for the residual $q$-Steiner system and the derived $q$-Steiner system result in two structures whose union has the same size as the related $q$-Steiner system, which is not the case for the definition given in [@KiLa15] and other possible definitions. This makes the residual $q$-Fano plane obtained by our construction to be a design which is almost as close as possible to a $q$-Fano plane. This definition of residual $q$-Steiner system and the construction of the residual $q$-Fano plane is a new direction for a research to solve the existence question of $q$-Steiner systems in general and $q$-Fano planes in particular.

The rest of this paper is organized as follows. In Section \[sec:residual\] we present a definition for a residual $q$-Steiner system, explain why this definition represents the appropriate $q$-analog definition, and compare it to the other definitions. In Section \[sec:comb\_struct\] a few combinatorial structures which are used in the construction are defined and some of their properties are discussed. In Section \[sec:represent\] we will discuss representation of subspaces for our construction. In Section \[sec:extend\] it will be explained how to extend and expand the subspaces in 