Let $q$ be a power of prime number $p$ and let $F\_{q^2}$ be a finite field of order $q^2$.
Suppose that "-" be a conjugation operation that is defined as follow:


$-:F\_{q^2} ‎\longrightarrow‎ ‎F\_{q^2}$


$x ‎\longmapsto‎ x^q$


Let $C$ be a cyclic code of length n over $F\_{q^2}$ with the generator polynomial $g(x)$ and let $\bar{C}=\lbrace \bar{c} : c \in C \rbrace$ be the conjugate code of $C$.


It is obvious that $\bar{C}$ is also a cyclic code. 


Is it possible to determine the generator polynomial of $\bar{C}$ from $g(x)$?



The generator polynomial of cyclic code
The generator polynomial of $\bar{C}$ is $‎\overline{g(x)}‎$. Because the conjugation operation is distributive over summation and multiplication. 


