N-dimensional Berlekamp-Massey algorithm for multiple arrays and construction of multivariate polynomials with preassigned zeros

Abstract

In this paper we propose an algorithm of finding a minimal set of linear recurring relations for a given finite set of n-dimensional arrays. This algorithm is an n-dimensional extension of the Berlekamp-Massey algorithm for multisequences as well as an extension of the n-dimensional Berlekamp-Massey algorithm for a single array. Our algorithm is used to obtain Groebner bases of ideals defined by preassigned zeros. The latter problem is an extension of that treated by Moeller and Buchberger in the sense that the zeros can be over any finite extension $\tilde K$ of the base field K. Our approach gives an efficient method of obtaining Groebner bases of ideals defined by zeros to construct n-dimensional cyclic codes (i.e. Abelian codes). In case that the dimension n is small, the computational complexity is of order O((ILd)²), where I, L and d are the degree of the extension of $\tilde K$ over K, the number of the zeros and the size of the independent point set for the Groebner basis, respectively.