Define a cyclic code. Show that there is a bijection between the cyclic codes of length $n$, and the factors of $X^{n}-1$ in $\mathbb{F}_{2}[X]$.

If $n$ is an odd integer then we can find a finite extension $K$ of $\mathbb{F}_{2}$ that contains a primitive $n$th root of unity $\alpha$. Show that a cyclic code of length $n$ with defining set $\left\{\alpha, \alpha^{2}, \ldots, \alpha^{\delta-1}\right\}$ has minimum distance at least $\delta$. Show that if $n=7$ and $\delta=3$ then we obtain Hamming's original code.

[You may quote a formula for the Vandermonde determinant without proof.]