On Rational-Interpolation Based List-Decoding and List-Decoding Binary Goppa Codes

TL;DR

This paper extends list-decoding algorithms for GRS and Goppa codes using algebraic techniques, achieving decoding up to the Johnson radius and revealing connections between different decoding algorithms.

Contribution

It introduces a novel fast interpolation polynomial construction and applies Wu's list decoder to binary Goppa codes, establishing links with the Guruswami-Sudan algorithm.

Findings

Decoding Goppa codes up to the Johnson radius

Establishing a connection between Wu and Guruswami-Sudan algorithms

Improved interpolation polynomial construction method

Abstract

We derive the Wu list-decoding algorithm for Generalised Reed-Solomon (GRS) codes by using Gr\"obner bases over modules and the Euclidean algorithm (EA) as the initial algorithm instead of the Berlekamp-Massey algorithm (BMA). We present a novel method for constructing the interpolation polynomial fast. We give a new application of the Wu list decoder by decoding irreducible binary Goppa codes up to the binary Johnson radius. Finally, we point out a connection between the governing equations of the Wu algorithm and the Guruswami-Sudan algorithm (GSA), immediately leading to equality in the decoding range and a duality in the choice of parameters needed for decoding, both in the case of GRS codes and in the case of Goppa codes.