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

Peter  Beelen; Tom Høholdt; Johan Sebastian Rosenkilde Nielsen; Yingquan Wu

doi:10.1109/TIT.2013.2243800

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

Peter Beelen, Tom Høholdt, Johan Sebastian Rosenkilde Nielsen, Yingquan Wu

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Abstract

We derive the Wu list-decoding algorithm for generalized Reed–Solomon (GRS) codes by using Gröbner bases over modules and the Euclidean algorithm as the initial algorithm instead of the Berlekamp–Massey algorithm. 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, 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.

Original language	English
Journal	I E E E Transactions on Information Theory
Volume	59
Issue number	6
Pages (from-to)	3269-3281
ISSN	0018-9448
DOIs	https://doi.org/10.1109/TIT.2013.2243800
Publication status	Published - 2013

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Copyright 2015 IEEE. IEEE. Personal use of this material is permitted. Permission
from IEEE must be obtained for all other uses, in any current or future media, including reprinting /republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

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title = "On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes",

abstract = "We derive the Wu list-decoding algorithm for generalized Reed–Solomon (GRS) codes by using Gr{\"o}bner bases over modules and the Euclidean algorithm as the initial algorithm instead of the Berlekamp–Massey algorithm. 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, 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.",

author = "Peter Beelen and Tom H{\o}holdt and Nielsen, {Johan Sebastian Rosenkilde} and Yingquan Wu",

note = "Copyright 2015 IEEE. IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting /republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.",

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AU - Beelen, Peter

AU - Høholdt, Tom

AU - Nielsen, Johan Sebastian Rosenkilde

AU - Wu, Yingquan

N1 - Copyright 2015 IEEE. IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting /republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

PY - 2013

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N2 - We derive the Wu list-decoding algorithm for generalized Reed–Solomon (GRS) codes by using Gröbner bases over modules and the Euclidean algorithm as the initial algorithm instead of the Berlekamp–Massey algorithm. 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, 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.

AB - We derive the Wu list-decoding algorithm for generalized Reed–Solomon (GRS) codes by using Gröbner bases over modules and the Euclidean algorithm as the initial algorithm instead of the Berlekamp–Massey algorithm. 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, 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.

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DO - 10.1109/TIT.2013.2243800

M3 - Journal article

VL - 59

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JO - I E E E Transactions on Information Theory

JF - I E E E Transactions on Information Theory

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