Class of boolean functions constructed using significant bits of linear recurrences over the ring $\mathbb{Z}_{2^n}$

In this work a class of functions is studied, which are built with the help of significant bits sequences on the ring $\mathbb{Z}_{2^n}$. This class is built with use of a function $\psi~: \mathbb{Z}_{2^n} \rightarrow \mathbb{Z}_2$. In public literature there are works in which $\psi$ is a linear function. Here we will use a non-linear $\psi$ function for this set.
It is known that the period of a polynomial $F$ in the ring $\mathbb{Z}_{2^n}$ is equal to $T(\mod 2)2^\alpha$, where $\alpha \in \overline {0, n-1}$. The polynomials for which it is true that $T(F) = T(F \mod 2)$, in other words $\alpha = 0$, are called marked polynomials. For our class we are going to use a polynomial with a maximum period as the characteristic polyomial.
In the present work we show the bounds of the given class: non-linearity, the weight of the functions, the Hamming distance between functions. The Hamming distance between these functions and functions of other known classes is also given.