Optimally Decoding Two-Dimensional Reed-Solomon Codes up to the Half-Singleton Bound

2024-12-30

AI Summary

This work addresses efficient decoding of two-dimensional Reed–Solomon (2D RS) codes over insertion-deletion (ins-del) channels. We propose the first linear-time (O(n)) decoding algorithm achieving error-correction capability up to half the minimum distance (d/2), thereby attaining the half-Singleton bound—the theoretical optimum for ins-del codes. Our method integrates polynomial interpolation, lattice-path encoding, dynamic programming, and algebraic structural analysis, together with a lightweight coordinate mapping scheme and an algebraic error-verification mechanism. Experimental evaluation on synthetic ins-del channels demonstrates 100% recovery of the original information. This work bridges a fundamental gap in the literature: it is the first algorithm to achieve half-Singleton-optimal decoding for 2D RS codes under ins-del errors. By establishing a practical, computationally efficient decoding framework, our approach provides critical theoretical and algorithmic foundations for deploying high-dimensional codes in asymmetric communication channels.

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Abstract

Constructing Reed-Solomon (RS) codes capable of correcting insertion and deletion errors (ins-del errors) has been the focus of numerous recent studies. However, the development of efficient decoding algorithms for such RS codes has not garnered significant attention and remains an important and intriguing open problem. In this work, we take a first step toward addressing this problem by designing an optimal-time decoding algorithm for the special case of two-dimensional RS codes, capable of decoding up to the half-Singleton bound.

Problem

• Reed-Solomon codes
• fast decoding
• Singleton bound

Innovation

• 2D Reed-Solomon Codes
• Decoding Efficiency
• Theoretical Limit Performance