I am writing a research paper where I have to describe Cyclic Codes briefly. So, here is how I describe it:
A linear code $\mathcal{C}$ in $\mathbb{F}^n_q$ is said to be cyclic if for every codeword $(c_0,\ c_1,\ ...,\ c_{n-1})\in\mathcal{C}$, the word $(c_{n-1},\ c_0,\ c_1,\ ...,\ c_{n-2})$ is also in $\mathcal{C}$. A codeword $(c_0,\ c_1,\ ...,\ c_{n-1})\in\mathcal{C}$ can be represented in a polynomial form as $c_0+ c_1 x +...+c_{n-1}x^{n-1}$. A linear code $\mathcal{C}$ in $\mathbb{F}^n_q$ is cyclic if and only if $\mathcal{C}$ is an ideal in residue class ring $\mathbb{F}_q[x]/(x^n-1)$. And the lowest degree polynomial $g(x)$ $( g(x)\neq 0)$ in that ideal is called the generator polynomial of $\mathcal{C}$. The generator polynomial $g(x)$ always divides $x^n-1$.
$\textbf{I am writing a journal, and I don't want to get my paper rejected because of a silly mistake.}$ $\textbf{Therefore I want your opinion on my brief description of cyclic codes.}$ $\textbf{Please correct me if there is anything wrong with my description of cyclic codes}.$
Some people might prefer to distinguish between code $C$ defined as a vector subspace of $\mathbb F_q^n$ and code $C(x)$ as a collection of polynomials that are an ideal in the residue class ring $\mathbb F_q^n[x]/(x^n-1)$. Also, you might (or might not) want to insist on $n$ and $q$ being relatively prime; the theory of cyclic codes is somewhat messier when the gcd of $n$ and $q$ is greater than $1$. Finally, when $q > 2$, many people prefer to define the generator polynomial as the (nonzero) monic polynomial of lowest degree in $C(x)$, and to point out that all the codeword polynomials are polynomial multiples of $g(x),$ in addition to pointing out that $g(x)$ is a divisor of $x^n-1$.