Reconstructing Reed-Solomon Codes from Multiple Noisy Channel Outputs

2026-01-15

AI Summary

This work addresses the problem of reconstructing a Reed–Solomon codeword from K independent noisy observations received over a q-ary symmetric discrete memoryless channel. The authors adapt the Koetter–Vardy soft-decision decoding algorithm to this multi-output setting and establish, for the first time, an explicit rate threshold that depends solely on the channel error probability \( p \) and the number of observations \( K \). They prove that when the code rate lies below this threshold and both the block length and alphabet size are sufficiently large, the reconstruction error probability can be made arbitrarily small. By integrating algebraic coding theory with soft information fusion, this study provides both a theoretically sound guarantee and a practical algorithm for efficient and reliable codeword reconstruction.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Applications

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication setting in which a sender transmits a codeword and the receiver observes K independent noisy versions of this codeword. In this work, we study the problem of efficient reconstruction when each of the $K$ outputs is corrupted by a $q$-ary discrete memoryless symmetric (DMS) substitution channel with substitution probability $p$. Focusing on Reed-Solomon (RS) codes, we adapt the Koetter-Vardy soft-decision decoding algorithm to obtain an efficient reconstruction algorithm. For sufficiently large blocklength and alphabet size, we derive an explicit rate threshold, depending only on $(p, K)$, such that the transmitted codeword can be reconstructed with arbitrarily small probability of error whenever the code rate $R$ lies below this threshold.

Problem

• sequence reconstruction
• Reed-Solomon codes
• noisy channel outputs
• discrete memoryless symmetric channel
• substitution errors

Innovation

• Reed-Solomon codes
• sequence reconstruction
• soft-decision decoding
• Koetter-Vardy algorithm
• discrete memoryless symmetric channel