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

In this paper, we study a class of functions built with the help of significant bits sequences on the ring $ \mathbb{Z}_{2 ^ n} $. This class is built with the use of a function $ \psi: \mathbb{Z}_{2 ^ n} \rightarrow \mathbb{Z}_2$. In public literature, there are results for a linear function $ \psi $. Here, we use a non-linear $ \psi $ function for this set. The period of a polynomial $F$ in the ring $ \mathbb{Z}_{2^n} $ is equal to $ T(F \bmod 2)2^{\alpha} $, where $ \alpha \in \{0,\ldots, n-1\} $. The polynomials for which $ T(F) = T(F \bmod 2) $, i.e. $ \alpha = 0 $, are called marked polynomials. For our class, we use a marked polynomial of the maximum period. 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.