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

Thomas Honold, Michael Kiermaier

TL;DR

This work investigates putative $q$-analogues of the Fano plane by studying sets of $3$-dimensional subspaces in $\\mathbb{F}_q^7$ that cover each $2$-dimensional subspace exactly once, connecting the problem to plane subspace codes in $\\mathrm{PG}(6,\\mathbb{F}_q)$. It develops a counting framework via spectra, constructs large plane codes using lifted Gabidulin (LMRD) codes and expurgation-augmentation techniques, and achieves a general code of size $q^8+q^5+q^4-q-1$ that can be extended to larger sizes but falls short of a true $q$-analogue due to collision phenomena. The paper provides a computer-free binary construction yielding $329$ planes and a computer-assisted ternary construction yielding $6977$, along with a refined analysis of extension steps and symmetry-invariant designs. Although the $q$-analogue remains unresolved, the work significantly advances subspace-code constructions, clarifies obstructions, and outlines concrete pathways for future exploration, including potential generalization to larger ambient dimensions $v$.

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.

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

TL;DR

This work investigates putative -analogues of the Fano plane by studying sets of -dimensional subspaces in that cover each -dimensional subspace exactly once, connecting the problem to plane subspace codes in . It develops a counting framework via spectra, constructs large plane codes using lifted Gabidulin (LMRD) codes and expurgation-augmentation techniques, and achieves a general code of size that can be extended to larger sizes but falls short of a true -analogue due to collision phenomena. The paper provides a computer-free binary construction yielding planes and a computer-assisted ternary construction yielding , along with a refined analysis of extension steps and symmetry-invariant designs. Although the -analogue remains unresolved, the work significantly advances subspace-code constructions, clarifies obstructions, and outlines concrete pathways for future exploration, including potential generalization to larger ambient dimensions .

Abstract

A set of -dimensional subspaces of , the -dimensional vector space over the finite field , is said to form a -analogue of the Fano plane if every -dimensional subspace of is contained in precisely one member of . The existence problem for such -analogues remains unsolved for every single value of . Here we report on an attempt to construct such -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 -analogue. In particular we find a ternary subspace code of new record size , and we are able to construct a binary subspace code of the largest currently known size in an entirely computer-free manner.

Paper Structure

This paper contains 7 sections, 14 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Counting Preliminaries
  3. Augmented LMRD Codes
  4. First Expurgating and Then Augmenting
  5. An Attempt to Construct a $q$-Analogue and its Failure
  6. Extensions
  7. Conclusion

Key Result

Lemma 1

Let $\mathcal{C}$ be a plane subspace code of size $M$ in $\mathop{\mathrm{PG}}\nolimits(6,\mathbb{F}_q)$ and $S$ any solid in $\mathop{\mathrm{PG}}\nolimits(6,\mathbb{F}_q)$. The spectrum $\alpha=\alpha(S)$ of $\mathcal{C}$ with respect to $S$ satisfies $\alpha_0+\alpha_1+\alpha_2+\alpha_3=M$ and t

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • proof
  • Theorem 1
  • ...and 22 more