for $\mathscr{C}$. Thus there is a matrix $T_t\in GL(k,q)$ such that $H= T_t\cdot H'$. This gives $$\begin{aligned}
(P_1)_t&=T_t(\alpha^{-1}\Theta((P_n)_t))=(\alpha^{-1}(P_n)\Theta\circ T)_t\\
(P_2)_t&=T_t\Theta((P_1)_t)=((P_1)\Theta\circ T)_t\\
(P_3)_t&=T_t\Theta((P_2)_t)=((P_2)\Theta\circ T)_t=((P_1)(\Theta\circ T)^2)_t\\
&~~\vdots\\
(P_n)_t&=T_t\Theta((P_n)_t)=((P_{n-1})\Theta\circ T)_t=((P_1)(\Theta\circ T)^{n-1})_t\end{aligned}$$

Furthermore, we have $$P_1=\alpha^{-1}(P_n)(\Theta\circ T)\Rightarrow P_1=\alpha^{-1}(P_1)(\Theta\circ T)^n
\Rightarrow \alpha P_1 = P_1(\Theta\circ T )^n$$ So, by putting $\tau :=\Theta\circ T$ and $P:=P_1$, we obtain that $$H=[P_t, (P\tau )_t, (P\tau ^2)_t,\dots,(P\tau
^{n-1})_t]$$ with $P\tau^n=\alpha P$.

“$\Leftarrow$” Let $\mathscr{C}$ be a code with parity check matrix $$H=[P_t, (P\tau )_t, (P\tau ^2)_t,\dots,(P\tau ^{n-1})_t]$$ with $P\in\F_q^k$, $\tau=\Theta\circ  T$ and $T\in GL(k,q)$ such that $P\tau^n=\alpha P$. Then, for any $\vec{c}=(c_0,c_1,\dots,c_{n-1})
 \in\mathscr{C}$ we have $\vec{c}H_t=\vec{0}$. This implies $$\begin{aligned}
 \sum_{i=0}^{n-1}c_i(P\tau^i)=\vec{0}&\Longrightarrow \left( \sum_{i=0}^{n-1}c_i(P\tau^i)\right)\tau=(\vec{0})\tau\\
&\Longrightarrow \sum_{i=0}^{n-2}\theta(c_i)(P\tau^{i+1})+\theta(c_{n-1})(P\tau^n)=\vec{0}\\
&\Longrightarrow \sum_{i=0}^{n-2}\theta(c_i)(P\tau^{i+1})+\theta(c_{n-1})(\alpha P)=\vec{0}\\
&\Longrightarrow (\alpha \theta(c_{n-1}),\theta(c_0),\dots,\theta(c_{n-2}))H_t=\vec{0},\end{aligned}$$ i.e. $(\alpha
\theta(c_{n-1}),\theta(c_0),\dots,\theta(c_{n-2}))\in\mathscr{C}$ and $\mathscr{C}$ is a skew $\alpha$-cyclic code.

\[GMaruta:3\] Let $g(x)=\sum_{i=0}^k a_ix^i$ with $a_k=1$ be a skew polynomial of degree $k$ in $\F_q[x;\theta]$ that divides on the right $x^n-\alpha$, where $\alpha\in\F_q^\ast$. Then $\mathscr{C}\subset\F_q^n$ is a skew $\alpha$-cyclic $[n,n-k]$-code over $\F_q$ with generator polynomial $g(x)$ if and only if $\mathscr{C}$ is a code with partity check matrix $$[P_t,(P\tau)_t,(P\tau^2)_t,\dots,(P\tau^{n-1})_t],$$ where $P=(1,0,\dots,0)$, $\tau=\Theta \circ T_g$ and $T_g$ is the companion matrix of $g(x)$, i.e. $$T_g=\begin{pmatrix}
0&1&0&\cdots &0\\
0&0&1&\cdots &0\\
\vdots&\vdots&\vdots&\ddots &\vdots\\
0&0&0&\cdots&1\\
-a_0&-a_1&-a_2&\cdots&-a_{k-1}\\
\end{pmatrix}.$$

Consider the following linear map $$\label{cod:pi4}
