to the non-commutative case (Theorem \[GMaruta:QT\]) and secondly, by using a factorization algorithm of A. Leroy in the commutative case, in Section \[Sec3\] we give two lower bounds for the minimum Hamming distance $d$ of skew $\theta$-module codes, in particular for skew constacyclic codes (Theorems \[corchete:2\] and \[corchete:log\]). Finally, as an application of the above results, we write five MAGMA Programs [@magma] and for small fields we construct a table with some $1$-generator skew QT codes (see Definition \[skew QT codes\]) which reach the known best minimum Hamming distance.

Properties of skew constacyclic codes and their duals {#Sec2}
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Along all this note, we will use the following notation.

Let $\F_q$ be a finite field with $q$ elements, where $q=p^n$ for some prime $p$. Define $\F_q^\ast:=\F_q-\{ 0\}$ and take $\alpha\in\F_q^\ast$. Let $\theta$ be an automorphism of $\F_q$ and let us recall here the definition of the main object which we will treat in this note.

\[skew GCC\] A linear code $\mathscr{C}\subseteq \mathbb{F}_q^n$ is called a *skew $(\alpha ,\theta )$-cyclic code* (simply a *skew $\alpha$-cyclic code*, or a $\theta$-constacyclic code) if $\mathscr{C}$ is invariant under the semi-linear map $$\phi_{\alpha,\theta} : (c_0,c_1\dots,c_{n-1})\mapsto (\alpha \theta(c_{n-1}),\theta(c_0),\dots,\theta(c_{n-2})).$$ Moreover, for some fixed $\theta$, we call a 