An Interpolation Procedure for List Decoding Reed--Solomon codes Based on Generalized Key Equations

TL;DR

This paper introduces a novel interpolation procedure linking syndrome-based and list decoding methods for Reed-Solomon codes, enabling decoding beyond half the minimum distance with a structured linear system approach.

Contribution

It reformulates Guruswami-Sudan interpolation conditions as Key Equations, leading to an efficient linear algebraic decoding algorithm for Reed-Solomon codes.

Findings

Provides a structured Block-Hankel linear system for decoding

Adapts the Fundamental Iterative Algorithm for solution

Achieves decoding beyond half the minimum distance

Abstract

The key step of syndrome-based decoding of Reed-Solomon codes up to half the minimum distance is to solve the so-called Key Equation. List decoding algorithms, capable of decoding beyond half the minimum distance, are based on interpolation and factorization of multivariate polynomials. This article provides a link between syndrome-based decoding approaches based on Key Equations and the interpolation-based list decoding algorithms of Guruswami and Sudan for Reed-Solomon codes. The original interpolation conditions of Guruswami and Sudan for Reed-Solomon codes are reformulated in terms of a set of Key Equations. These equations provide a structured homogeneous linear system of equations of Block-Hankel form, that can be solved by an adaption of the Fundamental Iterative Algorithm. For an $(n,k)$ Reed-Solomon code, a multiplicity $s$ and a list size $\listl$, our algorithm has time complexity \ON{\listl s^4n^2}.