We show that the known list-decoding algorithms for univariate multiplicity and folded Reed-Solomon codes can be made to run in $\tilde{O}(n)$ time. 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 list-decoding tasks in $\tilde{O}(n)$, where $n$ is the block-length of the code. Our algorithms have two main components. The first component builds upon the lattice-based approach of Alekhnovich (IEEE Trans. Inf. Theory 2005), who designed a $\tilde{O}(n)$ time list-decoding algorithm for Reed-Solomon codes approaching the Johnson radius. As part of the second component, we design $\tilde{O}(n)$ time algorithms for two natural algebraic problems: given a $(m+2)$-variate polynomial $Q(x,y_0,\dots,y_m) = \tilde{Q}(x) + \sum_{i=0}^m Q_i(x)\cdot y_i$ the first algorithm solves order-$m$ linear differential equations of the form $Q\left(x, f(x), \frac{df}{dx}, \dots,\frac{d^m f}{dx^m}\right) \equiv 0$ while 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 $m$ is an arbitrary constant and $\gamma$ is a field element of sufficiently high order. These algorithms can be viewed as generalizations of classical $\tilde{O}(n)$ time 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：我们证明了单变量重数码和折叠Reed-Solomon码的已知列表解码算法可在$\tilde{O}(n)$时间内运行。单变量重数码与FRS码是Reed-Solomon码的自然变体，因其在列表解码中的应用而被发现并研究。已知对于任意$\epsilon>0$和码率$r \in (0,1)$，存在显式构造的码族，这些码具有码率$r$，并能在多项式时间内从占比$(1-r-\epsilon)$的错误中进行列表解码，且列表大小恒定（Guruswami & Wang (IEEE Trans. Inform. Theory 2013) 及 Kopparty, Ron-Zewi, Saraf & Wootters (SIAM J. Comput. 2023)）。本文提出随机算法，可在$\tilde{O}(n)$时间内完成上述列表解码任务，其中$n$为码的块长。我们的算法包含两个主要组成部分：第一部分基于Alekhnovich (IEEE Trans. Inf. Theory 2005)的格方法，该方法设计了针对Reed-Solomon码的$\tilde{O}(n)$时间列表解码算法，可逼近Johnson半径。作为第二部分，我们为两个自然代数问题设计了$\tilde{O}(n)$时间算法：给定一个$(m+2)$元多项式$Q(x,y_0,\dots,y_m) = \tilde{Q}(x) + \sum_{i=0}^m Q_i(x)\cdot y_i$，第一个算法解形如$Q\left(x, f(x), \frac{df}{dx}, \dots,\frac{d^m f}{dx^m}\right) \equiv 0$的$m$阶线性微分方程；第二个算法解形如$Q\left(x, f(x), f(\gamma x), \dots,f(\gamma^m x)\right) \equiv 0$的函数方程，其中$m$为任意常数，$\gamma$为阶数足够高的域元素。这些算法可视为Sieveking (Computing 1972) 和 Kung (Numer. Math. 1974) 关于计算幂级数模逆的经典$\tilde{O}(n)$时间算法的推广，并可能具有独立的研究价值。