Methods of construction of skew linear recurrent sequences with maximal period based on the Galois polynomials factorization in the ring of matrix polynomials

Let $p$ be a prime, $R=\mathrm{GR}(q^d,p^d)$ be a Galois ring of cardinality $q^d$ and characteristic $p^d$, where $q=p^r$, $S=\mathrm{GR}(q^{nd},p^d)$ be an $R$-extension of degree $n$ and $\check{S}$ be an endomorphism ring of the module $_RS$. A sequence $v$ over $S$ with the recursion law
$$ \forall i\in\mathbb{N}_0 :\;\;\;v(i+m)= \\psi_{m-1}(v(i+m-1))+...+\psi_0(v(i)),\;\;\;\psi_0,...,\psi_{m-1}\in \check{S},$$
is called a skew LRS over $S$ with a characteristic polynomial $\Psi(x) = x^m - \sum_{j=0}^{m-1}\psi_jx^j$. The maximal period $T(v)$ of such sequence equals $\tau = (q^{mn}-1)p^{d-1}$. In this article we propose some new methods for construction the polynomials $\Psi(x)$, which define the recursion laws of skew linear recurrent sequences of maximal period. These methods are based on the search in $\check{S}[x]$ the divisors for classic Galois polynomials of period $\tau$ over $R$.