Grobner bases and alternant codes over Galois rings

Abstract

We give a new algorithm for the solution of the Hamming metric decoding problem for alternant codes over a Galois ring R. First we develop a comprehensive theory of Grobner bases over R|x/sub 1/,...,x/sub n/|, which is of independent interest. By specialising to the case of one variable, we show that the solution of the key equation can be determined as a certain minimal element in a Grobner basis of the solution module.