Extension of the Berlekamp-Massey algorithm to N dimensions

Abstract

We present an algorithm for finding a minimal set of linear recurring relations which are valid for a given n -dimensional array over any field, where the “minimality” is defined with respect to the partial order over the n -dimensional lattice. The algorithm is an extension of our two-dimensional version of the Berlekamp-Massey algorithm to more than two dimensions. The n -dimensional theory is based on more general concepts which can be reduced into those of the two-dimensional theory in the previous paper. In a typical case, the resulting set of polynomials characterizing the minimal linear recurring relations proves to be a Groebner basis of the ideal defined by the array, and consequently the structure of an n -dimensional linear feedback shift register with the minimum number of storage devices which can generate the array is determined by it.