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Representability of the direct sum of uniform q-matroids

Gianira N. Alfarano, Relinde Jurrius, Alessandro Neri, Ferdinando Zullo

Abstract

There are many similarities between the theories of matroids and $q$-matroids. However, when dealing with the direct sum of $q$-matroids many differences arise. Most notably, it has recently been shown that the direct sum of representable $q$-matroids is not necessarily representable. In this work, we focus on the direct sum of uniform $q$-matroids. Using algebraic and geometric tools, together with the notion of cyclic flats of $q$-matroids, we show that this is always representable, by providing a representation over a sufficiently large field.

Representability of the direct sum of uniform q-matroids

Abstract

There are many similarities between the theories of matroids and -matroids. However, when dealing with the direct sum of -matroids many differences arise. Most notably, it has recently been shown that the direct sum of representable -matroids is not necessarily representable. In this work, we focus on the direct sum of uniform -matroids. Using algebraic and geometric tools, together with the notion of cyclic flats of -matroids, we show that this is always representable, by providing a representation over a sufficiently large field.
Paper Structure (12 sections, 21 theorems, 91 equations)

This paper contains 12 sections, 21 theorems, 91 equations.

Table of Contents

  1. Our contribution:
  2. Outline of the paper:
  3. Preliminaries
  4. Rank-metric codes and q-systems
  5. q-Matroids
  6. The direct sum of q-matroids
  7. The direct sum of uniform q-matroids
  8. A representation of the direct sum of uniform q-matroids
  9. The particular case of uniform q-matroids of rank 1
  10. Necessary conditions for the representability
  11. Construction for two summands over small extension fields
  12. Conclusions and open questions

Key Result

Theorem 1.3

Let $\mathcal{C}$ be an $[n,k]_{q^m/q}$ non-degenerate rank-metric code and let $\mathcal{S}$ be the $\mathbb F_q$-span of the columns of a generator matrix $G$. Consider the isomorphism For every $u\in\mathbb F_{q^m}^k$ we have that

Theorems & Definitions (56)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: neri2021geometry
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • Theorem 1.11: alfarano2022cyclic
  • ...and 46 more