A construction of skew LRS of maximal period over finite fields based on the defining tuples of factors

Let $p$ be a prime number, $R=\mathrm{GF}(q)$ be a field of $q=p^r$ elements and $S=\mathrm{GF}(q^n)$ be an extension of $R$. Let $\breve S$ be the ring of all linear transformations of the space $_RS$. A linear recurrent sequence $v$ of order $m$ over the module $_{\breve S}S$ is said to be a skew linear recurrence sequence (skew LRS) of order $m$ over $S$. The period $T(v)$ of such sequence satisfies the inequality $T(v)\leq\tau=q^{mn}-1$. If $T(v)=\tau$ we call $v$ a skew LRS of maximal period (skew MP LRS). Here new classes of skew MP LRS based on the notion of the defining tuples of factors are constructed.