Combinatorial Structures for the Construction {#sec:comb_struct}
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The construction of the residual $q$-Fano plane given in the Section \[sec:construction\] will make use of a few combinatorial structures which are defined, described, and discussed in this section.

The first object is a *1-spread* (spread in short) in ${{\mathbb{F}}}_q^n$, where $n$ is even. A spread $S$ in ${{\mathbb{F}}}_q^n$ is a set of 2-subspaces of ${{\mathbb{F}}}_q^n$, such that each nonzero vector of ${{\mathbb{F}}}_q^n$ is contained in exactly one 2-subspace of $S$. It is well known that such a spread exists whenever $n$ is even.

A *1-parallelism* (parallelism in short) in ${{\mathbb{F}}}_q^n$ is a partition of the 2-subspaces of ${{\mathbb{F}}}_q^n$ into pairwise disjoint spreads. The number of 2-subspaces in such a spread is $\frac{q^n-1}{q^2-1}$. It was proved by Beutelspacher [@Beu74] that such a parallelism exists whenever $n$ is a power of 2.

We will be interested in a parallelism in ${{\mathbb{F}}}_q^4$, i.e. a partition of the $(q^2+q+1)(q^2+1)$ 2-subspaces of ${{\mathbb{F}}}_q^4$ into $q^2 +q+1$ disjoint spreads.

We further partition, for our construction of a residual $q$-Fano plane, the $q^2+q+1$ pairwise disjoint spreads of any given parallelism into three sets ${{\cal A}}$, ${{\cal B}}$, and ${{\cal C}}$. The set ${{\cal A}}$ contains one spread. The set 