Description of the feedback function of a nonlinear shift register

We consider a nonlinear shift register with a feedback function $F$ such that its period coincides with the period of some linear shift register. For this nonlinear shift register we study the methods of construction of a balanced mapping such that its coordinate functions are equivalent to the superposition of the binary function $f$ of $n$ variables and the transformation $\rho_l$ implemented by the shift register with the feedback function $l$. For a concrete function $f$ of the nonlinearity degree $3$ a polynomial of the function $F$ is obtained and its degree is calculated.