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.

翻译：我们证明，定义在任意有限域$\mathbb{F}$上，维度为$k$、块长度为$n$的Reed-Solomon码，可以从协议一致性$\sqrt{(k-1)n}$出发，在$\text{poly}(n, \log |\mathbb{F}|)$时间内实现确定性列表解码。在此之前，从Sudan和Guruswami-Sudan的著名成果开始的Reed-Solomon码列表解码算法，要么是时间复杂度为$\text{poly}(n, \log |\mathbb{F}|)$的随机算法，要么是时间复杂度多项式依赖于底层域特征的确定性算法。特别地，在素域$\mathbb{F}$上，此前尚无已知的能在$\text{poly}(n, \log |\mathbb{F}|)$时间内运行的确定性算法解决此问题。我们的主要技术要素是一个确定性算法，用于求解Sudan和Guruswami-Sudan算法中出现的二元多项式分解实例，该算法在任意有限域$\mathbb{F}$上的时间复杂度对域大小的依赖仅为$\text{poly}(\log |\mathbb{F}|)$。尽管即使在次数为$2$的一元多项式情况下，获取有限域上多项式分解的高效确定性算法也是一个基本未解难题，但我们证明，可以利用来自接收词额的额外信息，为列表解码Reed-Solomon码过程中出现的此类实例获得这样的算法。