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TitleGröbner bases over Galois rings with an application to decoding alternant codes
Author(s) Eimear Byrne, Patrick Fitzpatrick
TypeArticle in Journal
AbstractWe develop a theory of Gröbner bases over Galois rings, following the usual formulation for Gröbner bases over finite fields. Our treatment includes a division algorithm, a characterization of Gröbner bases, and an extension of Buchberger's algorithm. One application is towards the problem of decoding alternant codes over Galois rings. To this end we consider the module M = {(a, b) :aS equiv b mod x^r} of all solutions to the so-called key equation for alternant codes, where S is a syndrome polynomial. In decoding, a particular solution (Sum, Omega) in M is sought satisfying certain conditions, and such a solution can be found in a Gröbner basis of M. Applying techniques introduced in the first part of this paper, we give an algorithm which returns the required solution.
Length20
ISSN0747-7171
CopyrightAcademic Press
File
URL dx.doi.org/10.1006/jsco.2001.0442
LanguageEnglish
JournalJournal of Symbolic Computation
Volume31
Number5
Pages565-584
PublisherAcademic Press, Inc.
AddressDuluth, MN, USA
Year2001
Edition0
Translation No
Refereed No
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