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Short rank-metric codes and scattered subspaces

Stefano Lia, Giovanni Longobardi, Giuseppe Marino, Rocco Trombetti

TL;DR

The paper addresses constructing short minimal rank-metric codes ${[m+2,3,m-2]}_{q^m/q}$ by exploiting a geometric bridge between scattered ${\rm F}_q$-subspaces of ${\rm F}_{q^m}^3$ and linear cutting blocking sets. It introduces a parametric family ${\rm U}_\sigma$ of $(m+2)$-dimensional scattered subspaces in ${\rm F}_{q^m}^3$ via $\sigma$-linearized polynomials, and derives explicit scattering conditions from the root structure of a projected polynomial $Q(X)=X^{\sigma^2+1}-X^{\sigma+1}-X^{\sigma}+X$. For $m=5$ and $m=7$, the authors provide concrete arithmetic criteria on $(q,m)$ under which ${\rm U}_\sigma$ is maximum scattered, and they establish a 1-1 correspondence with cutting blocking sets, yielding linear sets in ${\rm PG}(2,q^m)$ with exactly three line-intersection characters. Consequently, the associated minimal rank-metric codes achieve the best possible minimum distance and a maximal second generalized weight, while duals exhibit rank-saturating properties with radius 2. The results extend the catalog of explicit minimal codes and open avenues for broader families and odd km cases.

Abstract

By exploiting the connection between scattered $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^3$ and minimal non degenerate $3$-dimensional rank metric codes of $\mathbb{F}_{q^m}^{n}$, $n \geq m+2$, described in [2], we will exhibit a new class of codes with parameters $[m+2,3,m-2]_{q^m/q}$ for infinite values of $q$ and $m \geq 5$ odd. Moreover, by studying the geometric structures of these scattered subspaces, we determine the rank weight distribution of the associated codes.

Short rank-metric codes and scattered subspaces

TL;DR

The paper addresses constructing short minimal rank-metric codes by exploiting a geometric bridge between scattered -subspaces of and linear cutting blocking sets. It introduces a parametric family of -dimensional scattered subspaces in via -linearized polynomials, and derives explicit scattering conditions from the root structure of a projected polynomial . For and , the authors provide concrete arithmetic criteria on under which is maximum scattered, and they establish a 1-1 correspondence with cutting blocking sets, yielding linear sets in with exactly three line-intersection characters. Consequently, the associated minimal rank-metric codes achieve the best possible minimum distance and a maximal second generalized weight, while duals exhibit rank-saturating properties with radius 2. The results extend the catalog of explicit minimal codes and open avenues for broader families and odd km cases.

Abstract

By exploiting the connection between scattered -subspaces of and minimal non degenerate -dimensional rank metric codes of , , described in [2], we will exhibit a new class of codes with parameters for infinite values of and odd. Moreover, by studying the geometric structures of these scattered subspaces, we determine the rank weight distribution of the associated codes.
Paper Structure (12 sections, 19 theorems, 86 equations)

This paper contains 12 sections, 19 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $q$-Systems, evasive and cutting subspaces
  4. Construction of a class of systems, $m$ odd
  5. Case
  6. Case
  7. Equivalence issue for $\mathcal{U}_{\sigma}$
  8. Linear sets associated with the cutting blocking sets $\mathcal{U}_{\sigma}$
  9. Characters of $L_{\mathcal{U}_{\sigma}}$ with respect to the lines of $PG(2,q^m)$
  10. Some consequences of previous results in terms of minimal codes
  11. Some consequences of previous results in terms of rank saturating systems
  12. Conclusions

Key Result

Theorem 2.1

The maps $\Phi$ and $\Psi$ are well-defined and they are the inverses of each other. Hence, they define a 1-to-1 correspondence between equivalence classes of $[n,k,d]_{q^m/q}$ codes and equivalence classes of $[n,k,d]_{q^m/q}$ systems. Moreover, under this correspondence that associates a $[n,k,d]_

Theorems & Definitions (28)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • Theorem 3.5
  • ...and 18 more