On minimal polynomials over Fqm and over Fq of a finite-length sequence over Fqm

In this paper we consider two problems, finding minimal polynomials over Fqm and over Fq of a finite-length sequence over Fqm, in a lattice and its sublattice respectively using lattice theory over polynomial rings. And we deduce a relationship between them by a lattice basis transformation matrix. As a byproduct, we present a new synthesis algorithm for the first problem just using operations in Fq instead of Fqm as in Berlekamp–Massey algorithm.