Then the columns of $[g^n]$ can be considered as points in $\P^{k-1}(\F_q)$ of an orbit of $\mathcal{T}$. Conversely, we can obtain similarly a skew constacyclic code from an orbit of a projective map of $\P^{k-1}(\F_q)$.

Now, consider $m$ orbits $\mathcal{O}_1,\mathcal{O}_2,\dots,\mathcal{O}_m$ of $\mathcal{T}$ with large $N$ and starting points $P_i\in\mathcal{O}_i$ for $i=1,...,m$. For simplicity, take $P_1\equiv P=(1,0,\dots,0)$ and define the matrix $$\begin{gathered}
 [P_t,(P\tau)_t,\dots,(P\tau^{n_1-1})_t;\
(P_2)_t,(P_2\tau)_t,\dots,\\ \dots, (P_2\tau^{n_2-1})_t;\
\cdots ;(P_m)_t,(P_m\tau)_t,\dots,(P_m\tau^{n_m-1})_t] \end{gathered}$$ by $[g^{n_1}]+P_2^{n_2}+\cdots+P_m^{n_m}$. Then, the matrix $[g^N]+P_2^N+\cdots+P_m^N$ generates a $SQT$ code whose generator $\boldsymbol{g}$ is given by the following result.

\[GMaruta:QT\] If $P_i\in(\F_q^{\theta})^k$ for $i=1,\dots, m$ are as above, then $[g^N]+P_2^N+\cdots+P_m^N$ generates a $1$-generator Skew Quasi-Twisted (SQT) $[mN,k]_q$-code with generator $$\boldsymbol{g}=\left(h^{\ast}(x),b_2(x^{-1})h^{\ast}(x),\dots,b_m(x^{-1})h^{\ast}(x)\right)\in R^N,$$ where $h^\ast(x)$ is as $h(x)$ in [*Proposition \[asterisco\]*]{} and $b_i(x)$ is the polynomial given by $(1,x,\dots,x^{k-1})P_i$ for $2\leq i\leq m$.

Define $H:=[g^N]+P_2^N+\cdots+P_m^N$ and note that $$\begin{gathered}
 H=[P_t,(P\tau)_t,\dots,(P\tau^{N-1})_t;\ (P_2)_t,(P_2\tau)_t,\dots,\\ \dots,(P_2\tau^{N-1})_t;\ \cdots ;\ (P_m)_t,(P_m\tau)_t,\dots,(P_m\tau^{N-1})_t] \end{gathered}$$ with $P_i\in(\F_q^{\theta})^k$, $\tau=\Theta\circ  T_g$ and $T_g\in GL(k,q)$ such that $P\tau^N=\alpha P$. By putting $H_i:=[(P_i)_t,(P_i\tau)_t,\dots,(P_i\tau^{N-1})_t]$ for all $i=1,\dots,n$, we get $H=[H_1|H_2|\cdots |H_m]$.

Take $P_1\equiv P:=(1,0,\dots,0)=\vec{e_1}$ and note that $P\tau^{j-1}=\vec{e_{j}}$ for all $j=1,\dots, k$, where $e_j$ is the $j$-th canonical vector of $\F_q^k$. Then, for $P_i\in(\F_q^{\theta})^k$ and $\lambda_{ij}\in \F_q^{\theta}$, we have $$P_i=\sum_{j=0}^{k-1}\lambda_{ij} P\tau^j=(P)\left(\sum_{j=0}^{k-1}\lambda_{ij} \tau^j\right)
=:P\cdot P_i(\tau),$$ where $P_i(x):=\sum_{j=0}^{k-1}\lambda_{ij}
x^j$. Thus 