lic codes in Theorem 2.1.

[**Theorem 2.1** ]{}[@Boucher1; @Boucher2] *Let $C$ be a skew cyclic code ($\sigma$-cyclic code) of length $n$ over $\mathbb{F}_q$ generated by a right divisor $g(x)=\sum_{i=0}^{n-k-1}g_ix^i+x^{n-k}$ of $x^n-1$. Then\
* *(i)  The generator matrix of $C$ is given by*

$$\left(
  \begin{array}{ccccccc}
    g_0 & \cdots & g_{n-k-1}& 1 & 0 & \cdots & 0\\
    0 & \sigma(g_0) & \cdots & \sigma(g_{n-k-1}) & 1 & \cdots & 0\\
    0 & \ddots & \ddots&  &   & \ddots& \vdots\\
    \vdots & & \ddots &\ddots &\cdots &\ddots &0\\
    0& \cdots & 0& \sigma^{k-1}(g_0) & \cdots & \sigma^{k-1}(g_{n-k-1})& 1\\
  \end{array}
\right)$$

*and $\mid C\mid=q^{n-{\rm deg}(g(x))}$.\
*

*(ii)  Let $x^n-1=h(x)g(x)$ and $h(x)=\sum_{i=0}^{k-1}h_ix^i$. Then $C^\perp$ is also a skew cyclic code of length $n$ generated by $\widetilde{h}(x)=x^{{\rm deg}(h(x))}\varphi(h(x))=1+\sigma(h_{k-1})x+\cdots+\sigma^k(h_0)x^k$, where $\varphi$ is an anti-automorphism of $\sigma$ defined as $\varphi(\sum_{i=0}^ta_ix^t)=\sum_{i=0}^tx^{-i}a_i$, where $ \sum_{i=0}^ta_ix^i\in R$. The generator matrix of $ C^\perp$ is given by* $$\left(
  \begin{array}{ccccccc}
    1 & \sigma(h_{k-1}) & \cdots& \sigma^k(h_0) & 0 & \cdots & 0\\
    0 & 1 & \sigma^2(h_{k-1}) & \cdots & \sigma^{k+1}(h_0) & \cdots & 0\\
    0 & 0 & \ddots&  &   & \ddots& 0\\
    \vdots & & \ddots &\ddots &\cdots &\ddots &\vdots\\
    0& \cdots & 0& 1 & \sigma^{n-k}(h_{k-1}) & \cdots& \sigma^{n-1}(h_0)\\
  \end{array}
\right)$$\
*and $\mid C^\perp\mid=q^k$.*\
*(iii)  For $c(x)\in R$, $c(x)\in C$ if and only if $c(x)h(x)=0$ in $R$.\
*

*(iv)  $C$ is a cyclic code of length $n$ over $\mathbb{F}_q$ if and only if the generator polynomial $g(x)\in \mathbb{F}_{p^d}[x]/(x^n-1)$.*

The monic polynomials $g(x)$ and $h(x)$ in Theorem 2.1 are called the *generator polynomial* and the *parity-check polynomial* of the skew cyclic code $C$, respectively.

[**Theorem 2.2** ]{} *Let $C$ be a skew cyclic code with the generator polynomial $g(x)$ and the check polynomial $h(x)$. Then 