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On the number of inequivalent linearized Reed-Solomon codes

Jonathan Mannaert, Marta Messia, Ferdinando Zullo

Abstract

Linearized Reed-Solomon (LRS) codes form an important family of maximum sum-rank distance (MSRD) codes that generalize both Reed--Solomon codes and Gabidulin codes. In this paper we study the equivalence problem for LRS codes and determine the number of inequivalent codes within this family. Using the correspondence between sum-rank metric codes and systems of $\mathbb{F}_q$-subspaces, we analyze the stabilizer of the Gabidulin system and derive a characterization of equivalence between LRS codes. In particular, we prove that two LRS codes are equivalent if and only if the sets of norms that define the codes coincide up to multiplication by an element of $\mathbb{F}_q^\ast$. This description allows us to reduce the classification problem to the action of $\mathbb{F}_q^\ast$ on subsets of $\mathbb{F}_q^\ast$. As a consequence, we derive formulas for the number of inequivalent linearized Reed-Solomon codes and illustrate the results with explicit examples.

On the number of inequivalent linearized Reed-Solomon codes

Abstract

Linearized Reed-Solomon (LRS) codes form an important family of maximum sum-rank distance (MSRD) codes that generalize both Reed--Solomon codes and Gabidulin codes. In this paper we study the equivalence problem for LRS codes and determine the number of inequivalent codes within this family. Using the correspondence between sum-rank metric codes and systems of -subspaces, we analyze the stabilizer of the Gabidulin system and derive a characterization of equivalence between LRS codes. In particular, we prove that two LRS codes are equivalent if and only if the sets of norms that define the codes coincide up to multiplication by an element of . This description allows us to reduce the classification problem to the action of on subsets of . As a consequence, we derive formulas for the number of inequivalent linearized Reed-Solomon codes and illustrate the results with explicit examples.
Paper Structure (14 sections, 27 theorems, 128 equations, 2 figures, 1 table)

This paper contains 14 sections, 27 theorems, 128 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Rank-metric codes and Gabidulin codes
  3. The vector framework
  4. Linearized polynomials
  5. The geometric framework
  6. Sum-rank metric codes and linearized Reed-Solomon codes
  7. Sum-rank metric codes
  8. Linearized Reed-Solomon codes
  9. The stabilizer of the Gabidulin system
  10. Equivalence classes of linearized Reed-Solomon codes
  11. Counting inequivalent Linearized Reed--Solomon codes
  12. A closed formula
  13. Examples
  14. Conclusions

Key Result

Theorem 2.2

Let $\phi:\mathbb{F}_{q^m}^n\rightarrow\mathbb{F}_{q^m}^n$ be an $\mathbb{F}_{q^m}$-linear isometry of $(\mathbb{F}_{q^m}^n,d_R)$, then there exist $\alpha\in\mathbb{F}_{q^m}^*$ and $A\in\mathop{\mathrm{GL}}\nolimits(n,q)$ such that $\phi(x)=\alpha xA$ for every $x\in\mathbb{F}_{q^m}^n$.

Figures (2)

  • Figure 1: Subgroup lattice of the cyclic group $C_{24}$. The highlighted portion corresponds to subgroups whose order divides $\gcd(24,12)=12$.
  • Figure 2: Subgroup lattice of the cyclic group $C_{6}$. The highlighted portion corresponds to subgroups whose order divides $\gcd(6,3)=3$.

Theorems & Definitions (57)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 47 more