$\alpha$-cyclic $[n,n-k]$-code. Finaly, since $g(x)$ is a monic polynomial which corresponds to the vector $\vec{g}\in\mathscr{C}$, we conclude that $\mathscr{C}$ is a skew $\alpha$-cyclic $[n,n-k]$-code over $\F_q$ with generator polynomial $g(x)$.\
“$\Rightarrow$” Let $\mathscr{C}$ be a skew $\alpha$-cyclic $[n,n-k]$-code over $\F_q$ with generator polynomial $g(x)=a_0+a_1x+\dots+a_{k-1}x^{k-1}+x^k$. Then the generator matrix of $\mathscr{C}$ in the canonical form is as in (\[MG\]). Let us show now that $H$ is the parity check matrix of $\mathscr{C}$. From $a_0P+a_1P\tau+\cdots+a_{k-1}P\tau^{k-1}+P\tau^k =\vec{0}$, it follows that $$\begin{aligned}
(a_0P+\cdots+P\tau^k )\tau=\vec{0}&\Leftrightarrow (\theta(a_0)P\tau+\theta(a_1)P\tau^2 +\cdots+P\tau^{k+1})=\vec{0}\\
(a_0P+\cdots+P\tau^k )\tau^2=\vec{0}&\Leftrightarrow (\theta^2(a_0)P\tau^2+\theta^2(a_1) P\tau^3+\cdots+P\tau^{k+2})=\vec{0}\\
\vdots\phantom{aaaaaaaaaa}&\phantom{aaaaaaaaaaaaaaa}{\vdots}\\
(a_0P+\cdots+P\tau^k
)\tau^{n-k-1}=\vec{0}&\Leftrightarrow
(\theta^{n-k-1}(a_0)P\tau^{n-k-1}+
\cdots+P\tau^{n-1})=\vec{0}.\end{aligned}$$ This implies that $G H_t=O$, i.e. $HG_t=O$ and $H$ is a parity check matrix of $\mathscr{C}$.

As an application of Theorem \[GMaruta:3\], the following MAGMA Program $4$ defines the function +PCM(n,g)+ which gives the parity check matrix in standard form of a skew a-cyclic codes in $\mathbb{F}_q^n$ with generator polynomial g :

\[Prog2\] 

    PCM:=function(n,g)
    V:=VectorSpace(F,n); g1:=R!g;
    if Degree(g1) ge 1 then
    d:=Degree(g1); CM:=VerticalJoin(HorizontalJoin( 
    ZeroMatrix(F,d-1,1), ScalarMatrix(F,d-1,1)), Matrix(F,1,d,
    [-Eltseq(g1)[k]: k in {1..d}])); P:=HorizontalJoin( 
    Matrix(F,1,1,[1]), ZeroMatrix(F,1,d-1)); TT:=R![0,1]; 
    PP:=P; PCM1:=P; PCM2:=P; for m in {1..n-1} do
    PCM1:=Matrix(F,1,d,[SpecialEvaluate(TT,PCM1[1][i]): 
    i in {1..d}])*CM; PCM2:=VerticalJoin(PCM2,PCM1); 
    end for; PCM3:=Transpose(PCM2); print 
    "==========================================="; print " ";
    print "Polynomial generator:"; print g1; print " ";
    print "Companion Matrix:"; print CM; print 