On the Root Finding Step in List Decoding of Folded Reed-Solomon Codes

TL;DR

This paper improves the root finding step in list decoding of folded Reed-Solomon codes by generalizing an algorithm, leading to better bounds, relaxed parameters, and introducing a new class of efficiently decodable codes.

Contribution

It introduces a multivariate generalization of the Roth-Ruckenstein algorithm for root finding, improving list decoding bounds and parameter flexibility for folded Reed-Solomon codes.

Findings

Enhanced list decoding bounds for folded Reed-Solomon codes.

Relaxed parameter constraints enabling more flexible code design.

Introduction of time-domain folded Reed-Solomon codes with efficient decoding.

Abstract

The root finding step of the Guruswami-Rudra list decoding algorithm for folded Reed-Solomon codes is considered. It is shown that a multivariate generalization of the Roth-Ruckenstein algorithm can be used to implement it. This leads to an improved bound on the size of the list produced by the decoder, as well as enables one to relax the constraints on the parameters of folded codes. Furthermore, the class of time-domain folded Reed-Solomon codes is introduced, which can be efficiently list decoded with the Guruswami-Rudra algorithm, and provides greater flexibility in parameter selection than the classical (frequency-domain) folded codes.