On the number of inequivalent linearized Reed-Solomon codes
This work addresses a classification problem in coding theory for researchers studying sum-rank metric codes, providing a theoretical framework to distinguish inequivalent LRS codes, which is incremental as it builds on known generalizations of Reed-Solomon and Gabidulin codes.
The paper tackled the problem of classifying equivalence for linearized Reed-Solomon (LRS) codes, which are maximum sum-rank distance codes, by proving that two LRS codes are equivalent if and only if their defining sets of norms coincide up to multiplication by an element of the finite field, and derived formulas to count the number of inequivalent codes.
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.