On Putative q-Analogues of the Fano Plane and RelatedCombinatorial Structures

A set〖 F〗_q of 3-dimensional subspaces of F_q^7, the 7-dimensional vector space over the finite field F_q^7, is said to form a q-analogue of the Fano plane if every 2-dimensional subspace of F_q^7 is contained in precisely one member of 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.