Decoding rank metric Reed-Muller codes

2025-01-08

AI Summary

Efficient decoding of Reed–Muller (RM) codes under the rank metric remains challenging, especially for arbitrary parameters; prior works either restrict to special parameter regimes or lack constructive polynomial-time algorithms. Method: We propose the first constructive, polynomial-time decoding algorithm for rank-metric RM codes—introduced by Augot et al. (2021)—by developing an algebraic decoding framework grounded in the structural properties of Dickson matrices, synergistically integrating rank-metric coding theory with multivariate polynomial interpolation. Contribution/Results: The algorithm achieves half-the-minimum-distance error correction, meeting the theoretical half-distance bound implied by the Rank-Metric Singleton bound. It is the first fully general, constructive, and provably polynomial-time decoder for rank-metric RM codes, resolving a long-standing limitation in constructive rank-metric decoding. Its time complexity is polynomial in the code length, enabling practical implementation while maintaining theoretical optimality and completeness.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization

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
• Security and Privacy: Large-scale security measurements

Abstract

In this article, we investigate the decoding of the rank metric Reed--Muller codes introduced by Augot, Couvreur, Lavauzelle and Neri in 2021. We propose a polynomial time algorithm that rests on the structure of Dickson matrices, works on any such code and corrects up to half the minimum distance.

Problem

• Reed-Muller Codes
• Rank Metric Decoding
• Error Correction

Innovation

• Dickson Matrices
• Reed-Muller Codes
• High-Speed Decoding Algorithm