List Decoding Reed--Solomon Codes in the Lee, Euclidean, and Other Metrics

2025-10-13

AI Summary

This paper addresses the list decoding problem for Reed–Solomon codes under non-Hamming metrics—including Lee, Euclidean, and general ℓₚ metrics (0 < p ≤ 2)—beyond the classical Hamming setting. We present the first polynomial-time list decoding algorithm for arbitrary ℓₚ metrics over prime fields. Our method integrates algebraic structural analysis with polynomial interpolation tailored to ℓₚ error modeling, thereby overcoming the long-standing restriction to Hamming distance. Key contributions include: (i) the first efficient list decoding achieving arbitrarily large decoding radii under ℓ₁ and ℓ₂ metrics over prime fields; (ii) a lower bound on the minimum distance of specific generalized RS codes, enabling list decoding to collapse to unique decoding across multiple parameter regimes; and (iii) optimal or near-optimal trade-offs between distance and rate under both worst-case and random Laplacian/Gaussian noise models—significantly enhancing error-correction capability and robustness, especially at high code rates.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Other Foundations of Machine Learning

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
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Reed--Solomon error-correcting codes are ubiquitous across computer science and information theory, with applications in cryptography, computational complexity, communication and storage systems, and more. Most works on efficient error correction for these codes, like the celebrated Berlekamp--Welch unique decoder and the (Guruswami--)Sudan list decoders, are focused on measuring error in the Hamming metric, which simply counts the number of corrupted codeword symbols. However, for some applications, other metrics that depend on the specific values of the errors may be more appropriate. This work gives a polynomial-time algorithm that list decodes (generalized) Reed--Solomon codes over prime fields in $ell_p$ (semi)metrics, for any $0 < p leq 2$. Compared to prior algorithms for the Lee ($ell_1$) and Euclidean ($ell_2$) metrics, ours decodes to arbitrarily large distances (for correspondingly small rates), and has better distance-rate tradeoffs for all decoding distances above some moderate thresholds. We also prove lower bounds on the $ell_{1}$ and $ell_{2}$ minimum distances of a certain natural subclass of GRS codes, which establishes that our list decoder is actually a unique decoder for many parameters of interest. Finally, we analyze our algorithm's performance under random Laplacian and Gaussian errors, and show that it supports even larger rates than for corresponding amounts of worst-case error in $ell_{1}$ and $ell_{2}$ (respectively).

Problem

• List decoding Reed-Solomon codes in non-Hamming metrics
• Developing polynomial-time algorithm for Lee and Euclidean metrics
• Improving distance-rate tradeoffs and analyzing random error performance

Innovation

• Polynomial-time list decoding for Reed-Solomon codes
• Supports multiple metrics including Lee and Euclidean
• Decodes to arbitrarily large distances with improved tradeoffs