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Most $q$-matroids are not representable

Sebastian Degen, Lukas Kühne

TL;DR

The paper proves that asymptotically almost all $q$-matroids are not representable. It develops a two-pronged approach: a coding-theoretic lower bound showing the total number of $q$-matroids $ \mathcal N_q(n)$ grows at least as $2^{q^{c n^2}}$, and a zero-pattern-based upper bound showing the number of representable $q$-matroids $ \mathcal R_q(n)$ grows at most like $q^{c' n^2}$ for some constants $c,c'$. Consequently, the ratio $ \mathcal R_q(n)/\u000a\mathcal N_q(n)$ tends to zero as $n\to\infty$, establishing a $q$-analogue of Nelson's theorem. The results hinge on linking constant-dimension codes to paving $q$-matroids and exploiting the algebraic structure of zero patterns in determinant-type polynomials. This provides a negative answer to Jurrius and Pellikaan's representability question, illuminating the landscape of $q$-matroids in coding-theoretic contexts.

Abstract

A $q$-matroid is the analogue of a matroid which arises by replacing the finite ground set of a matroid with a finite-dimensional vector space over a finite field. These $q$-matroids are motivated by coding theory as the representable $q$-matroids are the ones that stem from rank-metric codes. In this note, we establish a $q$-analogue of Nelson's theorem in matroid theory by proving that asymptotically almost all $q$-matroids are not representable. This answers a question about representable $q$-matroids by Jurrius and Pellikaan strongly in the negative.

Most $q$-matroids are not representable

TL;DR

The paper proves that asymptotically almost all -matroids are not representable. It develops a two-pronged approach: a coding-theoretic lower bound showing the total number of -matroids grows at least as , and a zero-pattern-based upper bound showing the number of representable -matroids grows at most like for some constants . Consequently, the ratio tends to zero as , establishing a -analogue of Nelson's theorem. The results hinge on linking constant-dimension codes to paving -matroids and exploiting the algebraic structure of zero patterns in determinant-type polynomials. This provides a negative answer to Jurrius and Pellikaan's representability question, illuminating the landscape of -matroids in coding-theoretic contexts.

Abstract

A -matroid is the analogue of a matroid which arises by replacing the finite ground set of a matroid with a finite-dimensional vector space over a finite field. These -matroids are motivated by coding theory as the representable -matroids are the ones that stem from rank-metric codes. In this note, we establish a -analogue of Nelson's theorem in matroid theory by proving that asymptotically almost all -matroids are not representable. This answers a question about representable -matroids by Jurrius and Pellikaan strongly in the negative.
Paper Structure (10 sections, 17 theorems, 41 equations)

This paper contains 10 sections, 17 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. $q$-Binomial coefficients
  5. $q$-Matroids
  6. Rank-metric codes and representable $q$-matroids
  7. A lower bound for the number of $q$-matroids
  8. An upper bound for the number of representable $q$-matroids
  9. Zero patterns and their connection to representable $q$-matroids
  10. The upper bound and the proof of the main result

Key Result

Theorem 1.2

Let $n$ be an integer, $\mathcal{R}_{q}(n)$ the number of representable $q$-matroids and $\mathcal{N}_{q}(n)$ the number of all $q$-matroids on ${\mathbb{F}}_q^n$, respectively. Then the ratio $\frac{\mathcal{R}_{q}(n)}{\mathcal{N}_{q}(n)}$ asymptotically vanishes, i.e.,

Theorems & Definitions (42)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • ...and 32 more