On putative q-Analogues of the Fano Plane and Related Combinatorial Structures

A set $\mathcal{F}_q$ of $3$-dimensional subspaces of $\mathbb{F}_q^7$, the $7$-dimensional vector space over the finite field $\mathbb{F}_q$, is said to form a $q$-analogue of the Fano plane if every $2$-dimensional subspace of $\mathbb{F}_q^7$ is contained in precisely one member of $\mathcal{F}_q$. The existence problem for such $q$-analogues remains unsolved for every single value of $q$. Here we report on an attempt to construct such $q$-analogues using ideas from the theory of subspace codes, which were introduced a few years ago by Koetter and Kschischang in their seminal work on error-correction for network coding. Our attempt eventually fails, but it produces the largest subspace codes known so far with the same parameters as a putative $q$-analogue. In particular we find a ternary subspace code of new record size $6977$, and we are able to construct a binary subspace code of the largest currently known size $329$ in an entirely computer-free manner.