Two-dimensional shift register synthesis and Gröbner bases for polynomial ideals over an integer residue ring

Abstract

In this paper we present an algorithm for finding a simplest n-dimensional linear feedback shift register which generates a given n-dimensional array over the integer residue ring Z m , where the term simplest means that the degree of every connection polynomial is minimal. This problem is an extension of the (one-dimensional) shift register synthesis over a field not only to n dimensions but also over the ring Z m . The result is useful for implementing encoders and decoders of Abelian codes over Z m .