Fast K\"otter-Nielsen-H{\o}holdt Interpolation in the Guruswami-Sudan Algorithm

TL;DR

This paper introduces a divide and conquer approach to accelerate the Kötter-Nielsen-Høholdt interpolation algorithm within the Guruswami-Sudan decoding process, achieving near-linear complexity and practical speed improvements.

Contribution

It presents a novel divide and conquer method that significantly speeds up the interpolation algorithm, making it more efficient for Reed-Solomon decoding.

Findings

Achieves asymptotic quasi-linear complexity in n.

Provides practical speed-up for classical algorithm implementations.

Enhances efficiency of Reed-Solomon decoding algorithms.

Abstract

The K\"otter-Nielsen-H{\o}holdt algorithm is a popular way to construct the bivariate interpolation polynomial in the Guruswami-Sudan decoding algorithm for Reed-Solomon codes. In this paper, we show how one can use Divide & Conquer techniques to provide an asymptotic speed-up of the algorithm, rendering its complexity quasi-linear in n. Several of our observations can also provide a practical speed-up to the classical version of the algorithm.