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

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.

37 pages, results were presented in part at Alcoma15

References in corpus (1)

• A New Approach to the Main Problem of Subspace Coding

Cited by in corpus (10)

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• A note on the linkage construction for constant dimension codes
• The Expurgation-Augmentation Method for Constructing Good Plane Subspace Codes
• Constructions and Bounds for Mixed-Dimension Subspace Codes
• Subspaces intersecting in at most a point
• Combinatorial Intricacies of Labeled Fano Planes
• On $\alpha$-points of $q$-analogs of the Fano plane
• A subspace code of size $333$ in the setting of a binary $q$-analog of the Fano plane
• New and Updated Semidefinite Programming Bounds for Subspace Codes