On some classes of nonlinear shift registers with the same cyclic structure

The paper is devoted to investigating the cyclic structure of an autonomous automaton $R(t)=R(G^n,\delta_f)$ named a shift register with feedback function $f$, $f\colon G^n\to G$, and transition function
$$ \delta_f(y_1,y_2,\dots,y_n)=(y_2,y_3,\dots,y_n,f(y_1,y_2,\dots,y_n)). $$
An important problem in this field of investigation consists of constructing a nonlinear automaton $R(f)$ of a given cyclic structure, in particular, possessing a cycle of length $2^n$ or $2^n-1$.