Parameters of Boolean functions generated by the most significant bits of linear recurrent sequences

The families of Boolean functions generated by the most significant bits of linear recurrent sequences over the ring $\mathbb Z_{2^n}$ with the marked characteristic polynomial are considered. For these families we investigate their cardinalities, the weights and the degree of nonlinearity of functions, distances between functions. We prove that there exists a rearrangement of arguments (one and the same for all functions in the family) such that the family contains functions which are close to some bent-functions.