Linear feedback shift registers and the minimal realization problem

Abstract

The Berlekamp–Massey algorithm solves the problem of finding the shortest linear feedback shift register which generates a given finite sequence of scalars. This problem is reinterpreted from the point of view of the realization theory and several extensions to sequences of matrices are analyzed. We give a generalization of the result on which the Berlekamp–Massey algorithm is based in terms of the partial Brunovsky indices of a sequence of matrices and propose an algorithm to obtain them for sequences of vectors. The results we obtain hold for arbitrary fields.