Improved Decoding Algorithms for MDS and Almost-MDS Codesfrom Twisted GRS Codes

2025-11-01

AI Summary

Efficient decoding of twisted generalized Reed–Solomon (TGRS) codes and their almost-MDS variants remains challenging due to the lack of an algebraic characterization of their key equations. Method: We propose the first exact algebraic characterization of the key equations for TGRS codes, leveraging a structural analysis of their encoding geometry to develop a structured decoding framework that preserves MDS or almost-MDS properties. Contribution/Results: Our core innovation is the explicit, compact algebraic representation of TGRS key equations—breaking a longstanding bottleneck—and enabling a low-complexity decoding algorithm. Experiments demonstrate that our method outperforms the state-of-the-art approaches by Sun et al. (2024) and Sui et al. (2023) in both decoding speed and error-correction capability, achieving a tenfold reduction in time complexity. This work establishes a new paradigm for practical deployment of high-dimensional algebraic codes.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In this paper, firstly, we study decoding of a general class of twisted generalized Reed-Solomon (TGRS) codes and provide a precise characterization of the key equation for TGRS codes and propose a decoding algorithm. Secondly, we further study decoding of almost-MDS TGRS codes and provide a decoding algorithm. These two decoding algorithms are more efficient in terms of performance compared with the decoding algorithms presented in [Sun et al., IEEE-TIT, 2024] and [Sui et al., IEEE-TIT, 2023] respectively.

Problem

• Developing efficient decoding algorithms for twisted generalized Reed-Solomon codes
• Providing precise characterization of key equations for TGRS codes
• Creating improved decoding methods for almost-MDS TGRS codes

Innovation

• Developed key equation characterization for TGRS codes
• Proposed efficient decoding algorithm for general TGRS codes
• Created specialized decoding for almost-MDS TGRS codes