1&\alpha-t&1\\
1&\alpha^2&\alpha-t\\
\end{pmatrix}
&,&&
m_{B_2}(t)&=det\begin{pmatrix}
-t & 0 & 1\\
1&\alpha^2-t&1\\
1&\alpha&\alpha^2-t\\
\end{pmatrix}\\
&=t^3+t+1 & &&&=t^3+t+1.\end{aligned}$$ This shows that there exist two different linear codes $\mathscr{C}_i:=Rg_i$, $i=1,2$, with the same associated minimal polynomial $m_{\tau_i}(x)$.

Finally, let us consider also $1$-generator skew quasi-twisted codes which can be easily defined from the notion of $1$-generator QT codes (see [@M2 $\S 1$]).

So, similarly to [@M2], we give first the following

\[skew QT codes\] Take $\alpha\in\F_q^\ast$ and let $\theta$ be an automorphism of $\F_q$. Denote by $R=\F_q[x;\theta]/(x^N-\alpha)$ the polynomial ring $\F_q[x;\theta]$ over $\F_q$ module $x^N-\alpha$. For $$\boldsymbol{g}=(g_1(x),g_2(x),\dots,g_m(x))\in R^N,$$ the set $$\boldsymbol{\mathscr{C}_g}=\{ (r(x)g_1(x),r(x)g_2(x),\dots,r(x)g_m(x)) \mid r(x)\in R \}$$ is called the *$1$-generator Skew Quasi-Twisted (SQT) code* with generator $\boldsymbol{g}$. Moreover, if $\alpha=1$ then $\boldsymbol{\mathscr{C}_g}$ is called the *$1$-generator Skew Quasi-Cyclic (SQC) code* with generator $\boldsymbol{g}$.

From Theorem \[GMaruta:3\] we know that an $[n,n-k]$-code over $\F_q$ is a skew $\alpha$-cyclic code with generator polynomial $g(x)=\sum_{i=0}^k a_ix^i$ with $a_k=1$ if and only if $\mathscr{C}$ is a linear code with parity check matrix $$[g^n]:=[P_t,(P\tau)_t,(P\tau^2)_t,\dots,(P\tau^{n-1})_t],$$ where $P=(1,0,\dots,0)$, $\tau=\Theta \circ T_g$ and $T_g$ is the companion matrix of $g(x)$. Now, let $\mathcal{T}$ be the projective map of $\P^{k-1}(\F_q)$ defined by $\tau$. We can say that $\mathcal{T}$ is defined by $g(x)$. 