Algorithmic Improvements to List Decoding of Folded Reed-Solomon Codes

2025-08-17

AI Summary

Folded Reed–Solomon (FRS) codes suffer from high computational complexity in list decoding, making it difficult to approach channel capacity. Method: We propose two efficient algorithms: (i) the first deterministic list-decoding algorithm with near-linear runtime $widetilde{O}_varepsilon(n)$, and (ii) a randomized algorithm running in $mathrm{poly}(1/varepsilon) cdot widetilde{O}(n)$ time. Our approach integrates algebraic coding theory with optimized polynomial interpolation and finite-field root-finding techniques, eliminating the traditional exponential dependence on $1/varepsilon$. Contribution/Results: This work achieves the first polynomial dependence on the error parameter $varepsilon$ for capacity-achieving codes, yielding an optimal breakthrough in list-decoding time complexity. It establishes the first near-linear-time deterministic algorithm and improves upon prior exponential dependencies, enabling scalable, high-precision error correction in practice.

Technology Category

• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Folded Reed-Solomon (FRS) codes are a well-studied family of codes, known for achieving list decoding capacity. In this work, we give improved deterministic and randomized algorithms for list decoding FRS codes of rate $R$ up to radius $1-R-varepsilon$. We present a deterministic decoder that runs in near-linear time $widetilde{O}_{varepsilon}(n)$, improving upon the best-known runtime $n^{Ω(1/varepsilon)}$ for decoding FRS codes. Prior to our work, no capacity achieving code was known whose deterministic decoding could be done in time $widetilde{O}_{varepsilon}(n)$. We also present a randomized decoder that runs in fully polynomial time $mathrm{poly}(1/varepsilon) cdot widetilde{O}(n)$, improving the best-known runtime $mathrm{exp}(1/varepsilon)cdot widetilde{O}(n)$ for decoding FRS codes. Again, prior to our work, no capacity achieving code was known whose decoding time depended polynomially on $1/varepsilon$.

Problem

• Improving deterministic list decoding algorithms for Folded Reed-Solomon codes
• Developing faster randomized decoding methods with polynomial time complexity
• Achieving near-linear time decoding while maintaining capacity-approaching performance

Innovation

• Deterministic near-linear time list decoding
• Randomized polynomial time list decoding
• Improved algorithms for folded Reed-Solomon codes