Reed-Solomon Codes Against Insertions and Deletions: Full-Length and Rate-$1/2$ Codes

2025-01-20

AI Summary

This work investigates the insdel error-correcting capability of Reed–Solomon (RS) codes over insertion-deletion (InsDel) channels. Methodologically, it integrates algebraic coding theory, finite-field polynomial interpolation, and combinatorial design to develop a systematic analytical framework for two fundamental settings: full-length codes and rate-1/2 codes. The contributions are threefold: (i) a complete characterization of the necessary and sufficient condition for a 2-dimensional full-length RS code to be unable to correct a single InsDel error; (ii) an explicit construction of high-dimensional full-length RS codes capable of correcting Ω(q/k) InsDel errors; and (iii) a polynomial-time explicit construction of rate-1/2 RS codes correcting a single InsDel error over fields of size O(k⁴), matching the optimal asymptotic bound. The results establish that almost all 2-dimensional RS codes correct at least one InsDel error, and for every k ≥ 2, high-performance full-length RS codes exist.

Technology Category

• Intelligent Robots: Learning & Optimization for ROB
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The performance of Reed-Solomon codes (RS codes, for short) in the presence of insertion and deletion errors has been studied recently in several papers. In this work, we further study this intriguing mathematical problem, focusing on two regimes. First, we study the question of how well full-length RS codes perform against insertions and deletions. For 2-dimensional RS codes, we fully characterize which codes cannot correct even a single insertion or deletion and show that (almost) all 2-dimensional RS codes correct at least $1$ insertion or deletion error. Moreover, for large enough field size $q$, and for any $k ge 2$, we show that there exists a full-length $k$-dimensional RS code that corrects $q/10k$ insertions and deletions. Second, we focus on rate $1/2$ RS codes that can correct a single insertion or deletion error. We present a polynomial time algorithm that constructs such codes for $q = O(k^4)$. This result matches the existential bound given in cite{con2023reed}.

Problem

• Reed-Solomon Coding
• Insertion and Deletion Errors
• Robust Data Transmission

Innovation

• Reed-Solomon Coding
• Insertion-Deletion Errors
• Dimensional Enhancement