Re-Encoding Techniques for Interpolation-Based Decoding of Reed-Solomon Codes

TL;DR

This paper explores re-encoding and periodicity projection techniques to improve the interpolation step in decoding Reed-Solomon codes using the Guruswami-Sudan algorithm, simplifying computations and enhancing decoding efficiency.

Contribution

It introduces the periodicity projection method and analyzes its benefits, including low Hamming weight and regular structure, for more efficient interpolation in Reed-Solomon decoding.

Findings

Periodicty projection reduces the complexity of the interpolation step.

Modified received vectors enable compression of the linear system.

Recovery of the interpolated polynomial is simplified with the periodicity projection.

Abstract

We consider interpolation-based decoding of Reed-Solomon codes using the Guruswami-Sudan algorithm (GSA) and investigate the effects of two modification techniques for received vectors, i.e., the re-encoding map and the newly introduced periodicity projection. After an analysis of the latter, we track the benefits (that is low Hamming weight and regular structure) of modified received vectors through the interpolation step of the GSA and show how the involved homogeneous linear system of equations can be compressed. We show that this compression as well as the recovery of the interpolated bivariate polynomial is particularly simple when the periodicity projection was applied.