Synthesizing all linearized shift-registers of the minimal or required length

An efficient algorithm synthesizing all q-linearized shift-registers of the minimal or required length generating a sequence of length N over a finite field F
_qm
is considered. This algorithm, which is a generalization of the Berlekamp-Massey algorithm, has time complexity O(N
²
) operations in IF
_qm
, and can be applied for efficient solving the key equation when decoding Gabidulin codes up to and beyond half the minimum rank distance .