Define a cyclic code of length $N$.

Show how codewords can be identified with polynomials in such a way that cyclic codes correspond to ideals in the polynomial ring with a suitably chosen multiplication rule.

Prove that any cyclic code $\mathcal{X}$ has a unique generator, i.e. a polynomial $c(X)$ of minimum degree, such that the code consists of the multiples of this polynomial. Prove that the rank of the code equals $N-\operatorname{deg} c(X)$, and show that $c(X)$ divides $X^{N}+1$.

Let $\mathcal{X}$ be a cyclic code. Set

$$\mathcal{X}^{\perp}=\left\{y=y_{1} \ldots y_{N}: \sum_{i=1}^{N} x_{i} y_{i}=0 \text { for all } x=x_{1} \ldots x_{N} \in \mathcal{X}\right\}$$

(the dual code). Prove that $\mathcal{X}^{\perp}$ is cyclic and establish how the generators of $\mathcal{X}$ and $\mathcal{X}^{\perp}$ are related to each other.

Show that the repetition and parity codes are cyclic, and determine their generators.