" ";
    print "Parity Check Matrix:"; print PCM3; print " ";
    print "===========================================";
    end if; return "end"; end function;

Consider the finite field $\F_4=\{ 0,1,w,w^2\}$, where $w^2+w+1=0$, and the Frobenius automorphism $\theta$ defined by $\theta(x)=x^2$ for any $x\in\F_4$. The skew $w$-cyclic $[7,3]_4$-code generate by the polynomial $g(x)=x^4+x^2+w^2 x+1$ has a generator matrix $$G:=\begin{pmatrix}
1 & w^2 & 1 & 0 &1 & 0 & 0 \\
0 & 1 & w & 1 &0 & 1 & 0 \\
0& 0 &1 & w^2 &1& 0 & 1 \\
\end{pmatrix}.$$ By the command +PCM(7,\[1,w\^2,1,0,1\])+ in Program $4$, we obtain that its parity check matrix is $H:=[P_t,(P\tau)_t,(P\tau^2)_t,\dots,(P\tau^{6})_t]$, where $P=(1,0,0,0)$, $\tau=\Theta \circ T_g$ and $T_g$ is the companion matrix of $g(x)$: $$T_g=\begin{pmatrix}
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
1& w^2&1&0 \\
\end{pmatrix}, \qquad H=\begin{pmatrix}
1&0&0&0 &1&0&1\\
0&1&0&0 &w^2&1&w^2\\
0&0&1&0 &1&w&0\\
0&0&0&1 &0&1&w^2\\
\end{pmatrix}.$$

A linear code $\mathscr{C}\subseteq \F_q^n$ with parity check matrix of the form $$[P_t, (P\tau)_t,
(P\tau^2)_t,\dots,(P\tau^{n-1})_t],$$ with $P\in\F_q^k$, $\tau=\Theta\circ T$ y $T\in GL(k,q)$ is called *a code $\mathscr{C}$ defined by $(\tau,P,n)$*. Furthermore, one can define also the following set $$\Gamma_k^\alpha :=\{ (\tau,P,n) \mid \exists \text{ a skew $\alpha$-cyclic $[n,n-k]_q$-code
defined by } (\tau,P,n) \}.$$

\[GMaruta:4\] Let $\mathscr{C}_i$ be the code defined by $(\tau_i,P_i,n)\in
