Fast list-decoding of univariate multiplicity and folded Reed-Solomon codes

TL;DR

The paper achieves nearly-linear-time ( randomized $ ilde{O}(n)$) list-decoding for univariate multiplicity and folded Reed-Solomon codes up to capacity, by marrying a lattice-based interpolation framework with fast solvers for differential and functional equations. The core idea is to replace the costly interpolation step with a shortest-vector computation in a carefully constructed polynomial lattice, then solve the resulting algebraic equations in nearly-linear time via Newton-style divide-and-conquer using a conjugate form $Q^{\dagger}$. Pruning to a constant-sized list, building on recent capacity-list-decoding results, yields a complete near-linear-time decoder for capacity-achieving regimes, while deterministic Johnson-radius decoding is also achieved. The methods rely on fast polynomial arithmetic, lattice reduction over $\mathbb{F}[x]$, and fast Hermite/functional interpolation, with potential independent interest in solving linear differential and functional equations efficiently.

Abstract

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 $ε>0$, and rate $r \in (0,1)$, there exist explicit families of these codes that have rate $r$ and can be list decoded from a $(1-r-ε)$ 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 $Q\left(x, f(x), f(γx), \dots,f(γ^m x)\right) \equiv 0$, where $m$ is an arbitrary constant and $γ$ 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.

Theorems & Definitions (78)

• definition 1.1: Folded Reed-Solomon Krachkovsky2003GuruswamiR2008
• definition 1.2: Multiplicity codes RosenbloomT1997Nielsen2001KoppartySY2014
• theorem 1.3
• theorem 1.4
• theorem 1.5
• theorem 1.6
• theorem 1.7
• theorem 2.1
• lemma 3.1: Taylor expansion of polynomials
• lemma 3.2