linear code $\mathscr{C}\subseteq \mathbb{F}_q^n$ a *skew constacyclic code* (or, a *skew pseudo-cyclic code*) if $\mathscr{C}$ is a skew $\alpha$-cyclic code for some $\alpha\in \F_q^\ast$.

The above definition can be reinterpreted in an algebraic way by the following

\[boucher:A\] Let $\alpha\in\F_q^\ast$. A linear code $\mathscr{C}\subseteq \mathbb{F}_q^n$ is a skew $(\alpha,\theta )$-cyclic code if and only if $\mathscr{C}$ is invariant under the semi-linear map $\T:=\Theta \circ A$, that is, $(\vec{c})\T\in \mathscr{C}$ for all $\vec{c}\in\mathscr{C}$, where $\Theta\colon \F_q^n\to\F_q^n$ is defined by $( (c_0,\dots,c_{n-1}))\Theta:=(\theta(c_0),\dots,\theta(c_{n-1}))$ and $A$ is the $n\times n$ matrix given by $$A:=\left( \begin{array}{ c | c c c } 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \\ \hline \alpha & 0 & \cdots & 0 \end{array}\right).$$

Let $\mathscr{C}\subseteq \mathbb{F}_q^n$ be a skew $(\alpha ,\theta )$-cyclic code. Then $$\mathscr{C}\star (\Theta\circ A )\subseteq \mathscr{C},$$ where $\mathscr{C}\star (\Theta\circ A):=\{ (\vec{c})\Theta A\mid \vec{c}\in\mathscr{C} \}$. When $\theta=Id$, a skew $(\alpha, Id)$-cyclic code is simply a constacyclic code.

From now on, we mainly show some algebraic and geometric properties for the dual code of a skew constacyclic codes.

First of all, with the purpose of giving an algebraic structure to skew $\alpha$-cyclic 