Grobner bases over Galois rings with an application to decoding alternant codes

We develop a theory of Grobner bases over Galois rings, following the usual formulation for Grobner bases over finite fields. Our treatment includes a division algorithm, a characterization of Grobner 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 = 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 (Sigma, Omega) is an element of M is sought satisfying certain conditions, and such a solution can be found in a Grobner basis of M. Applying techniques introduced in the first part of this paper, we give an algorithm which returns the required solution. (C) 2001 Academic Press.