Lower estimates of ranks of coordinate sequences of maximal period linear recurrent sequences over the non-trivial Galois ring

The Galois ring is called non-trivial if it is not a field and a residue ring. For the maximal period linear recurrent sequence over the non-trivial Galois ring $R=GR(q^n,p^n)$, $p\ge5$, new lower estimates of ranks of coordinate sequences with numbers $s$ such that $s=kr+2$, $r=\lg_pq$, $k\in\mathbb N_0$ are obtained.