Details:
Title  Gröbner bases over Galois rings with an application to decoding alternant codes  Author(s)  Eimear Byrne, Patrick Fitzpatrick  Type  Article in Journal  Abstract  We 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 socalled 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.  Length  20  ISSN  07477171  Copyright  Academic Press 
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 URL 
dx.doi.org/10.1006/jsco.2001.0442 
Language  English  Journal  Journal of Symbolic Computation  Volume  31  Number  5  Pages  565584  Publisher  Academic Press, Inc.  Address  Duluth, MN, USA  Year  2001  Edition  0  Translation 
No  Refereed 
No 
