Deterministic list decoding of Reed-Solomon codes
Soham Chatterjee, Prahladh Harsha, Mrinal Kumar
TL;DR
The paper resolves a longstanding open problem by giving the first fully deterministic algorithm that list-decodes Reed-Solomon codes from the Johnson radius in time poly$(n,\\log|\\mathbb{F}|)$ for any finite field $\\mathbb{F}$. It achieves this by replacing randomized polynomial factorization steps with a structured, deterministic factorization strategy tailored to the interpolation polynomials in Sudan and Guruswami–Sudan, centered on Hensel lifting rather than Newton iteration. A key technical innovation is the local splitting framework, which uses per-point information from the received word to partition and refine the bivariate factorization problem into coprime, manageable pieces, culminating in a stable set from which all close codewords can be recovered by interpolation. The approach demonstrates that the algebraic structure of coding-theoretic instances can bypass the general factorization barrier, offering a concrete derandomization blueprint for a core problem in coding theory. This advances both the theory of deterministic computation over finite fields and the practical toolkit for robust RS decoding in deterministic settings.
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 $\text{poly}(n, \log |\mathbb{F}|)$. 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 $\text{poly}(n, \log |\mathbb{F}|)$ 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 $\text{poly}(n, \log |\mathbb{F}|)$ 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 $\text{poly}(\log |\mathbb{F}|)$ 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.