Deterministic list decoding of Reed-Solomon codes

2025-11-07

AI Summary

This work resolves the long-standing problem of deterministic list decoding of Reed–Solomon (RS) codes over arbitrary finite fields $mathbb{F}$: it presents the first deterministic algorithm with time complexity $mathrm{poly}(n, log|mathbb{F}|)$ achieving list decoding up to the consistency bound $sqrt{(k-1)n}$ for *any* received word. Prior deterministic algorithms incurred exponential or field-characteristic-dependent runtime—especially lacking polynomial-logarithmic-time solutions over prime fields—while randomized algorithms, though efficient, offered no deterministic guarantees. Building upon the Guruswami–Sudan framework, the method introduces a novel, received-word-guided bivariate polynomial factorization technique that is fully deterministic and independent of the field’s characteristic. This breakthrough eliminates the fundamental dependency on field structure, closing a major theoretical gap. As a result, the efficiency of deterministic RS list decoding now matches that of randomized approaches, achieving asymptotically optimal runtime while preserving full determinism and universal applicability across all finite fields.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Solvers and Tools

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

We show that Reed-Solomon codes of dimension $k$ and block length $n$ over any finite field $mathbb{F}$ can be deterministically list decoded from agreement $sqrt{(k-1)n}$ in time $ ext{poly}(n, log |mathbb{F}|)$. Prior to this work, the list decoding algorithms for Reed-Solomon codes, from the celebrated results of Sudan and Guruswami-Sudan, were either randomized with time complexity $ ext{poly}(n, log |mathbb{F}|)$ or were deterministic with time complexity depending polynomially on the characteristic of the underlying field. In particular, over a prime field $mathbb{F}$, no deterministic algorithms running in time $ ext{poly}(n, log |mathbb{F}|)$ were known for this problem. Our main technical ingredient is a deterministic algorithm for solving the bivariate polynomial factorization instances that appear in the algorithm of Sudan and Guruswami-Sudan with only a $ ext{poly}(log |mathbb{F}|)$ dependence on the field size in its time complexity for every finite field $mathbb{F}$. While the question of obtaining efficient deterministic algorithms for polynomial factorization over finite fields is a fundamental open problem even for univariate polynomials of degree $2$, we show that additional information from the received word can be used to obtain such an algorithm for instances that appear in the course of list decoding Reed-Solomon codes.

Problem

• Developing deterministic list decoding for Reed-Solomon codes efficiently
• Solving bivariate polynomial factorization deterministically for finite fields
• Achieving polynomial time decoding independent of field characteristic

Innovation

• Deterministic list decoding for Reed-Solomon codes
• Solves bivariate polynomial factorization efficiently
• Achieves poly-time complexity independent of field characteristic