On the Hamming distance between binary representations of linear recurrent sequences over field $GF(2^k)$ and ring $\mathbb{Z}_{2^n}$

Linear recurrent sequences over the field $GF(2^k)$ and over the ring $\mathbb{Z}_{2^n}$ with dependent recurrent relations are considered. We establish the bounds for the Hamming distance between two binary sequences obtained from the initial sequences by replacing each element by its image under the action of arbitrary maps into the field of two elements.