Define a cyclic binary code of length $n$.

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

Prove that any cyclic binary code $C$ has a unique generator, that is, 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 $X^{n}-1$.

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