On the number of inequivalent linearized Reed-Solomon codes

TL;DR

This paper investigates the classification of linearized Reed-Solomon codes, providing formulas for counting inequivalent codes by analyzing their structural properties and the action of the multiplicative group on defining norms.

Contribution

It offers a complete characterization of code equivalence and derives explicit formulas for counting inequivalent LRS codes, advancing understanding of their structural diversity.

Findings

Two LRS codes are equivalent iff their defining norms differ by a scalar in _q^*

The classification reduces to the action of _q^* on subsets of _q^*

Derived formulas for the number of inequivalent LRS 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 $\mathbb{F}_q$-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.