On the number of inequivalent linearized Reed-Solomon codes

2026-03-18

AI Summary

This study addresses the classification of linearized Reed–Solomon (LRS) codes up to equivalence, aiming to determine the number of pairwise inequivalent codes within this family. By establishing a correspondence between LRS codes and subspace systems over finite fields, the authors provide the first necessary and sufficient condition for code equivalence: two LRS codes are equivalent if and only if their norm sets coincide under the action of the multiplicative group of the finite field. This characterization reduces the classification problem to counting orbits under a group action. Combining techniques from rank-metric coding theory, finite field algebra, and group actions, the paper derives an explicit formula for the number of inequivalent LRS codes and validates its correctness through concrete examples.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography

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.

Problem

• Linearized Reed-Solomon codes
• equivalence problem
• inequivalent codes
• sum-rank metric
• MSRD codes

Innovation

• linearized Reed-Solomon codes
• sum-rank metric
• code equivalence
• Gabidulin system
• inequivalent codes