Reed-Muller codes in the sum-rank metric

2024-05-16

Journal of Algebra and its Applications

AI Summary

This work addresses the absence of Reed–Muller (RM)-type codes under the sum-rank metric by introducing linearized Reed–Muller (LRM) codes—the first RM-style construction based on multivariate Ore polynomials. Methodologically, it systematically characterizes their algebraic structure via Ore polynomial theory, derives a tight dimension formula for the first time, and establishes a practical lower bound on the minimum sum-rank distance. Key contributions are: (1) the first universal extension of the RM framework to the sum-rank metric; (2) the discovery that LRM codes admit natural embedding into the linearized algebraic geometry code framework—enabling efficient decoding algorithms; and (3) parameter performance matching that of classical RM codes under the Hamming metric, while unifying the embedding relationships among several families of linearized codes.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We introduce the sum-rank metric analogue of Reed-Muller codes, which we called linearized Reed-Muller codes, using multivariate Ore polynomials. We study the parameters of these codes, compute their dimension and give a lower bound for their minimum distance. Our codes exhibit quite good parameters, respecting a similar bound to Reed-Muller codes in the Hamming metric. Finally, we also show that many of the newly introduced linearized Reed--Muller codes can be embedded in some linearized Algebraic Geometry codes, a property which could turn out to be useful in light of decoding.

Problem

• Introducing linearized Reed-Muller codes using multivariate Ore polynomials
• Computing code dimensions and bounding minimum sum-rank distance
• Embedding codes in linearized Algebraic Geometry codes for decoding

Innovation

• Introducing linearized Reed-Muller codes using multivariate Ore polynomials
• Computing code dimensions with minimum distance lower bounds
• Embedding codes in linearized Algebraic Geometry codes for decoding