We show that the known list-decoding algorithms for univariate multiplicity and folded Reed-Solomon (FRS) codes can be made to run in nearly-linear time. This yields, to our knowledge, the first known family of codes that can be decoded in nearly linear time, even as they approach the list decoding capacity. Univariate multiplicity codes and FRS codes are natural variants of Reed-Solomon codes that were discovered and studied for their applications to list-decoding. It is known that for every $\epsilon >0$, and rate $R \in (0,1)$, there exist explicit families of these codes that have rate $R$ and can be list-decoded from a $(1-R-\epsilon)$ fraction of errors with constant list size in polynomial time (Guruswami & Wang (IEEE Trans. Inform. Theory, 2013) and Kopparty, Ron-Zewi, Saraf & Wootters (SIAM J. Comput. 2023)). In this work, we present randomized algorithms that perform the above tasks in nearly linear time. Our algorithms have two main components. The first builds upon the lattice-based approach of Alekhnovich (IEEE Trans. Inf. Theory 2005), who designed a nearly linear time list-decoding algorithm for Reed-Solomon codes approaching the Johnson radius. As part of the second component, we design nearly-linear time algorithms for two natural algebraic problems. The first algorithm solves linear differential equations of the form $Q\left(x, f(x), \frac{df}{dx}, \dots,\frac{d^m f}{dx^m}\right) \equiv 0$ where $Q$ has the form $Q(x,y_0,\dots,y_m) = \tilde{Q}(x) + \sum_{i = 0}^m Q_i(x)\cdot y_i$. The second solves functional equations of the form $Q\left(x, f(x), f(\gamma x), \dots,f(\gamma^m x)\right) \equiv 0$ where $\gamma$ is a high-order field element. These algorithms can be viewed as generalizations of classical algorithms of Sieveking (Computing 1972) and Kung (Numer. Math. 1974) for computing the modular inverse of a power series, and might be of independent interest.

翻译：我们证明，已知的单变量重数码和折叠Reed-Solomon（FRS）码的列表译码算法可以实现在近线性时间内运行。据我们所知，这提供了第一族已知的即使接近列表译码容量也能在近线性时间内译码的码。单变量重数码和FRS码是Reed-Solomon码的自然变体，因其在列表译码中的应用而被发现和研究。已知对于任意$\epsilon > 0$和码率$R \in (0,1)$，存在这些码的显式构造族，其码率为$R$，并能在多项式时间内从$(1-R-\epsilon)$比例的差错中实现恒定列表大小的列表译码（Guruswami & Wang (IEEE Trans. Inform. Theory, 2013) 和 Kopparty, Ron-Zewi, Saraf & Wootters (SIAM J. Comput. 2023)）。在本文中，我们提出了在近线性时间内完成上述任务的随机算法。我们的算法包含两个主要部分。第一部分基于Alekhnovich (IEEE Trans. Inf. Theory 2005) 的格基方法，该方法设计了逼近Johnson半径的Reed-Solomon码的近线性时间列表译码算法。作为第二部分，我们为两个自然代数问题设计了近线性时间算法。第一个算法求解形如$Q\left(x, f(x), \frac{df}{dx}, \dots,\frac{d^m f}{dx^m}\right) \equiv 0$的线性微分方程，其中$Q$具有形式$Q(x,y_0,\dots,y_m) = \tilde{Q}(x) + \sum_{i = 0}^m Q_i(x)\cdot y_i$。第二个算法求解形如$Q\left(x, f(x), f(\gamma x), \dots,f(\gamma^m x)\right) \equiv 0$的函数方程，其中$\gamma$是一个高阶域元素。这些算法可视为Sieveking (Computing 1972) 和Kung (Numer. Math. 1974) 用于计算幂级数模逆的经典算法的推广，并可能具有独立的研究意义。