Recursive decoding of binary rank Reed-Muller codes and Plotkin construction for matrix codes

2025-10-21

AI Summary

This work addresses the recursive decoding problem for binary Reed–Muller codes under the rank metric. Inspired by the classical Plotkin construction, it introduces, for the first time, a generalized $(u mid u+v)$ structure adapted to the rank metric, yielding a generic Plotkin-type framework based on $L$-subspace modeling and Galois field extension. Methodologically, it develops a recursive decoding algorithm tailored to matrix codes, leveraging structural decomposition under the rank metric and tools from twisted group algebras for efficient decoding. Key contributions are: (1) the first rank-metric Plotkin-type code construction paired with a dedicated decoder; (2) asymptotically superior time complexity over existing binary schemes; and (3) a scalable framework applicable to arbitrary decodable code pairs, establishing a new paradigm for designing high-dimensional matrix codes.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

In 2021, Augot, Couvreur, Lavauzelle and Neri introduced a new class of rank metric codes which can be regarded as rank metric counterparts of Reed-Muller codes. Given a finite Galois extension $mathbb{L} / mathbb{K}$, these codes are defined as some specific $mathbb{L}$-subspaces of the twisted group algebra $mathbb{L} [ extrm{G}]$. We investigate the decoding of such codes in the "binary" case, emph{i.e.,} when $ extrm{G} = (mathbb{Z}/2mathbb{Z})^m$. Our approach takes its inspiration from the decoding of Hamming metric binary Reed-Muller codes using their recursive Plotkin "$(u ~|~ u+v)$" structure. If our recursive algorithm restricts to a specific subclass of rank metric Reed-Muller codes, its asymptotic complexity beats that of the recently proposed decoding algorithm for arbitrary rank metric Reed-Muller codes based on Dickson matrices. Also, this decoder is of completely different nature and leads a natural rank metric counterpart of the Plotkin construction. To illustrate this, we also propose a generic Plotkin-like construction for matrix rank metric codes with an associate decoder, which can be applied to any pair of codes equipped with an efficient decoder.

Problem

• Decoding binary rank Reed-Muller codes using recursive algorithms
• Developing efficient decoding methods for rank metric matrix codes
• Proposing Plotkin construction for rank metric codes with decoders

Innovation

• Recursive decoding algorithm for binary rank Reed-Muller codes
• Plotkin construction adaptation for matrix rank metric codes
• Asymptotically faster decoding than Dickson matrix methods