& w & 0 & w^2 & 0 & 1\\
\end{pmatrix}.$$ By the command +DualCode(8,w,\[1,0,w\^2,0,w,0,1\])+ in Program $2$ we see that the dual code $\mathscr{C}^\perp$ is the skew $w^2$-cyclic $[8,6,2]_4-$code ($w^2=w^{-1}$) generated by the polynomial $x^2+w^2$ and with generator matrix $$\begin{pmatrix}
1&0&w&0&0&0&0&0\\
0&1&0&w^2&0&0&0&0\\
0&0&1&0&w&0&0&0\\
0&0&0&1&0&w^2&0&0\\
0&0&0&0&1&0&w&0\\
0&0&0&0&0&1&0&w^2\\
\end{pmatrix}.$$

The following two results show a geometric property of the dual codes of skew constacyclic codes. More precisely, we will see that the columns of a parity check matrix of a skew constacyclic code can be considered as points in a projective space which are particular orbits under the action of the semi-linear map associated to the code. Therefore, in line with [@M1], in the non-commutative case we obtain the following two results.

\[GMaruta:2\] Let $\mathscr{C}$ be a linear $[n,n-k]$-code over $\F_q$. Then $\mathscr{C}$ is a skew $\alpha$-cyclic code if and only if $\mathscr{C}$ has a parity check matrix of the form $$[P_t, (P\tau )_t, (P\tau ^2)_t,\dots,(P\tau ^{n-1})_t]$$ such that $P\tau^n=\alpha P$, where $P\in\F_q^k$, $\tau=\Theta\circ T$ and $T\in GL(k,q)$.

“$\Rightarrow$” Let $\mathscr{C}$ be a skew $\alpha$-cyclic $[n,n-k]$-code over $\F_q$ with a parity check matrix $H=[(P_1)_t,(P_2)_t,\dots,(P_n)_t]$, where $(P_i)\in\F_q^k$. Then by Theorem \[GMaruta:1e\] we see that $H'=[\alpha^{-1}
\Theta((P_n)_t),\Theta((P_1)_t),\Theta((P_2)_t),\dots,\Theta((P_{n-1})_t)]$ is also a parity check matrix 