Periodic properties of a simplest 2-linear shift register

The state transition graph of a simplest self-controlled 2-linear shift register over Galois ring $R=GR(2^{rn},2^n)$ is studied. An upper bound for the length of a cycle in this graph is obtained. In the case $R=\mathbf Z_{2^n}$, states belonging to cycles of maximal length are described and the number of these states is evaluated.