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

TL;DR

This paper explores the challenging problem of constructing q-analogues of the Fano plane, reports on an unsuccessful attempt to create such structures, but achieves record sizes for related subspace codes.

Contribution

It introduces new large subspace codes related to the q-analogue of the Fano plane, including the largest known binary subspace code constructed without computers.

Findings

Largest known ternary subspace code with size 6977

Largest known binary subspace code with size 329

Failed to construct the q-analogue of the Fano plane

Abstract

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.