minimal polynomial may be associated with two different codes. In fact, if $\mathscr{C}_1=\mathscr{C}_2$ then $m_{\tau_1}=m_{\tau_2}$, but the reverse is not true in general, as the following example shows.

Consider $R:=\F_4[x;\theta]$, $g_1=1+wx+x^2+x^3$ and $g_2=1+w^2 x+x^2+x^3$ in $R$, where $\theta$ is the Frobenius automorphism. Let $\mathscr{C}_i:=Rg_i$ be a skew $1$-cyclic $[14,11]_4$-codes for $i=1,2$. Since $g_1(x)\not=
g_2(x)$ in $R/R(x^{14}-1)$, we have $\mathscr{C}_1\not=\mathscr{C}_2$. On the other hand, we get $$\begin{aligned}
\tau_1&=\Theta\circ\begin{pmatrix}
0 & 1 &0 \\
0&0&1\\
1&\alpha&1\\
\end{pmatrix}
&&,&
\tau_2&=\Theta\circ\begin{pmatrix}
0 & 1 &0 \\
0&0&1\\
1&\alpha^2&1\\
\end{pmatrix}
 \end{aligned}$$ with minimal polynomials $m_{\tau_1}(x)=m_{B_1}(x^2)=x^6+x^2+1$ and $m_{\tau_2}(x)=m_{B_2}(x^2)=x^6+x^2+1$ respectively (e.g., see [@TT2]), where $$\begin{aligned}
B_1&=\begin{pmatrix}
0 & 1 &0 \\
0&0&1\\
1&\alpha&1\\
\end{pmatrix}_\theta \begin{pmatrix}
0 & 1 &0 \\
0&0&1\\
1&\alpha&1\\
\end{pmatrix}
&,&&
B_2&=\begin{pmatrix}
0 & 1 &0 \\
0&0&1\\
1&\alpha^2&1\\
\end{pmatrix}_\theta \begin{pmatrix}
0 & 1 &0 \\
0&0&1\\
1&\alpha^2&1\\
\end{pmatrix} \\
&=\begin{pmatrix}
0 & 1 &0 \\
0&0&1\\
1&\alpha^2&1\\
\end{pmatrix} \begin{pmatrix}
0 & 1 &0 \\
0&0&1\\
1&\alpha&1\\
\end{pmatrix}
&&&
&=\begin{pmatrix}
0 & 1 &0 \\
0&0&1\\
1&\alpha&1\\
\end{pmatrix} \begin{pmatrix}
0 & 1 &0 \\
0&0&1\\
1&\alpha^2&1\\
\end{pmatrix} \\
&=\begin{pmatrix}
0 & 0 &1 \\
1&\alpha&1\\
1&\alpha^2&\alpha\\
\end{pmatrix}
&&&
&=\begin{pmatrix}
0 & 0 &1 \\
1&\alpha^2&1\\
1&\alpha&\alpha^2\\
\end{pmatrix}, \\
m_{B_1}(t)&=det\begin{pmatrix}
-t & 0 & 1\\
