Reconstructing Reed-Solomon Codes from Multiple Noisy Channel Outputs

TL;DR

This paper develops an efficient method for reconstructing Reed-Solomon codes from multiple noisy outputs, establishing a rate threshold for reliable decoding based on channel noise and number of observations.

Contribution

It adapts the Koetter-Vardy decoding algorithm for multiple noisy outputs and derives an explicit rate threshold for successful reconstruction.

Findings

Reconstruction is possible below a specific rate threshold depending on noise and observations.

The adapted algorithm achieves efficient decoding with arbitrarily small error probability.

Explicit rate thresholds are derived for large blocklengths and alphabet sizes.

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.