skew a-cyclic codes of $\mathbb{F}_q^n$ with $\theta (x):=x^{p^t}$:

\[Prog0\] 

    SD:=function(n,a)
    P:=[0]; for i in [1..n-2] do
    P:=P cat [0]; end for; T:= [-a] cat P cat [1]; f:=R!T; 
    V:=VectorSpace(F,n); dd:=[]; E:=[x : x in F | x ne 0]; 
    S:=CartesianProduct(E,CartesianPower(F,n-1));
    for s in S do
    ll:=[s[1]] cat [p : p in s[2]];
    if LeadingCoefficient(R!ll) eq 1 then
    q,r:=Quotrem(f,R!ll); if r eq R![0] then
    R!ll; dd := dd cat [R!ll]; end if; end if; end for;
    return dd; end function;

In $\mathbb{F}_4^{14}$ with $\theta (x):=x^{2}$, the command +SD(14,1)+ gives $603$ different nontrivial right divisors of $X^{14}-1$, i.e. $603$ different nontrivial skew cyclic codes of $\mathbb{F}_4^{14}$ instead of $25$ different nontrivial cyclic codes of $\mathbb{F}_4^{14}$ in the commutative case.

The following technical result will be useful to prove Theorem \[GMaruta:1e\].

\[Tironi\] Let $\mathscr{C}\subseteq \F_q^n$ be a linear code and let $\mathcal{T}\colon \F_q^n\to\F_q^n$ be a semi-linear map such that $\mathscr{C}\star \mathcal{T}\subseteq
\mathscr{C}$.

(a) If $\mathcal{T}$ is invertible, then $\mathscr{C}\star
    \mathcal{T}= \mathscr{C}$. Furthermore, $\mathscr{C}\star
    \T=\mathscr{C}$ if and only if $\mathscr{C}\star
    (\T)^{-1}=\mathscr{C}$;

(b) if $\T$ is as in [*Proposition \[boucher:A\]*]{}, then $\mathscr{C}\star
    \mathcal{T}=\mathscr{C}\star (\T)^{-1}=\mathscr{C}$; moreover, the dual code $\mathscr{C}^\perp$ is a linear code invariant under $\T'=\Theta^{-1}\circ ({}^t \! A)_{\theta^{-1}}$ with $\mathscr{C}^{\perp}\star
    \T'=\mathscr{C}^{\perp}\star (\T')^{-1}=\mathscr{C}^{\perp}$, where $M_{\theta^{-1}}:=[\theta^{-1}(m_{ij})]$ 