by $$h(x):=\theta^k(\hbar_0^{-1})\left[\sum_{i=0}^k \theta^i\left( \hbar_{k-i}\right)x^i\right]\ ,$$ where $\hbar(x):=\hbar_0+\hbar_1x+\cdots+\hbar_kx^k$ is such that $x^n-\theta^{-k}(\alpha)=g(x)\hbar(x).$

Since $\mathscr{C}$ is a skew $\alpha$-cyclic code, by Theorem \[GMaruta:1e\] we know that the dual code $\mathscr{C}^\perp$ is a skew $\alpha^{-1}$-cyclic code. Thus from Theorem \[GMaruta:1\] it follows that there exists a unique monic polynomial $h(x)$ of minimal degree in $R:=\F_q[x;\theta]$ such that $\mathscr{C}^\perp=Rh(x)$. Therefore by [@B2 Theorem 8] we deduce that there exist $\hbar(x)\in R$ and $c\in\mathbb{F}_q^*$ such that $$X^n-c=g(x)\hbar(x)$$ and $h(x)=\theta^k(\hbar_0^{-1})g^\perp(x)$, where $g^\perp(x):=\sum_{i=0}^k \theta^i\left( \hbar_{k-i}\right)x^i$ and $\hbar(x)=\hbar_0+\hbar_1x+\cdots+\hbar_kx^k$. Since $g(x)\in R$ is monic and $X^n-\alpha =t(x)g(x)$ for some monic $t(x)\in R$, by [@B1 Lemma 2] we obtain that $$X^n-\theta^{-k}(\alpha)=g(x)s(x)$$ for some $s(x)\in R$. Hence $c-\theta^{-k}(\alpha)=g(x)(s(x)-\hbar(x))$ and since $\deg g(x)\geq 1$, we conclude that $c=\theta^{-k}(\alpha)$.

The following MAGMA Program $3$ defines a function +DualCode(n,a,g)+ which gives all the main informations about the dual code of a skew a-cyclic codes of length n with generator polynomial g :

\[Prog1\] 

    DualCode:=function(n,a,g)
    k:=n-Degree(R!g); C:=a^(p^(k*(r-t))); P:=[0];
    for i in [1..n-2] do
    P:=P cat [0]; end for;
    T:= [-C] cat P cat [1]; f:=R!T; g1:=R!g; E:=[x : x in F | x ne 0];
    S:=CartesianProduct(E,CartesianPower(F,n-Degree(g1)));
    for ss in S do
    ll:=[ss[1]] cat [p : p in ss[2]];
    if g1*R!ll eq f then
    h:=R!ll; end 