Constructing pseudorandom sequences by means of $2$-linear shift register

We describe the periodicity properties for almost all $2$-nobreakdash-linear recurrent sequences generated by $2$-linear shift register with identical connection polynomials of maximal period. A class of self-control nonlinear functions are suggested such that the existence of maximally possible cycles in a transition graph of states is guaranteed. Linear output functions preserving the period of sequence are described.