Synthesis of two-dimensional linear feedback shift registers and Groebner bases

Abstract

In this paper we discuss about how to design a two-dimensional linear feedback shift register. It is a switching circuit capable of generating a prescribed doubly periodic array. In particular, it can be used as an encoder of a two-dimensional cyclic code. Our method is based on a two-dimensional extension of the Berlekamp-Massey algorithm for synthesis of a (one-dimensional) linear feedback shift register. In the sequel, we make clear that our problem is equivalent to constructing a Groebner basis of the ideal which is defined by the given array.