101 \\              011 \\              000 \\
                000                 110                 011                 101                         110                 000                 101                 011                         000                 110                 011                 101                         110                 000                 101                 011
           \end{array}$        \end{array}$        \end{array}$        \end{array}$                \end{array}$        \end{array}$        \end{array}$        \end{array}$                \end{array}$        \end{array}$        \end{array}$        \end{array}$                \end{array}$        \end{array}$        \end{array}$        \end{array}$
  ----- ------------------- ------------------- ------------------- ------------------- ------- ------------------- ------------------- ------------------- ------------------- ------- ------------------- ------------------- ------------------- ------------------- ------- ------------------- ------------------- ------------------- -------------------
  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

  : The extension space for $q=2$ ($C$ is the code and $C_i$, $i=1,2,3$, are its cosets)[]{data-label="tab:extension"}

The construction of the residual $q$-Fano plane will start from sets of subspaces from ${{\mathbb{F}}}_q^4$. The subspaces of these sets will be extended in various ways to 3-subspaces of ${{\mathbb{F}}}_q^6$, in a way that all these extensions will result in the the residual $q$-Fano plane. The extension space will have an important role in these extensions as will be explained in Sections \[sec:represent\] and \[sec:construction\]. The methods in which subspaces are extended is explained in Section \[sec:represent\].

We end this section with a connection between the subspaces of ${{\cal A}}$ and the subspaces of the set ${{\cal D}}$.

\[lem:q+1\] A 2-subspace $X$ of ${{\mathbb{F}}}_q^4$ can be expanded in $q+1$ distinct ways to a 3-subspace of ${{\mathbb{F}}}_q^4$.

A 2-subspace $X$ has $q+1$ pairwise linearly independent vectors. 