Algorithmic Improvements to List Decoding of Folded Reed-Solomon Codes

TL;DR

This paper introduces significantly faster deterministic and randomized algorithms for list decoding Folded Reed-Solomon codes, achieving near-linear and polynomial time complexities respectively, thus advancing decoding efficiency at capacity.

Contribution

It provides the first near-linear time deterministic decoder and polynomial time randomized decoder for capacity-achieving FRS codes, improving over previous exponential and super-polynomial runtimes.

Findings

Deterministic decoder runs in near-linear time $ ilde{O}_ ext{epsilon}(n)$.

Randomized decoder runs in polynomial time $ ext{poly}(1/ ext{epsilon}) imes ilde{O}(n)$.

Both algorithms achieve decoding up to the list decoding capacity.

Abstract

Folded Reed-Solomon (FRS) codes are a well-studied family of codes, known for achieving list decoding capacity. In this work, we give improved deterministic and randomized algorithms for list decoding FRS codes of rate $R$ up to radius $1-R-\varepsilon$. We present a deterministic decoder that runs in near-linear time $\widetilde{O}_{\varepsilon}(n)$, improving upon the best-known runtime $n^{\Omega(1/\varepsilon)}$ for decoding FRS codes. Prior to our work, no capacity achieving code was known whose deterministic decoding could be done in time $\widetilde{O}_{\varepsilon}(n)$. We also present a randomized decoder that runs in fully polynomial time $\mathrm{poly}(1/\varepsilon) \cdot \widetilde{O}(n)$, improving the best-known runtime $\mathrm{exp}(1/\varepsilon)\cdot \widetilde{O}(n)$ for decoding FRS codes. Again, prior to our work, no capacity achieving code was known whose decoding time depended polynomially on $1/\varepsilon$. Our results are based on improved pruning procedures for finding the list of codewords inside a constant-dimensional affine subspace.