(a) Define what it means to say that $C$ is a binary cyclic code. Explain the bijection between the set of binary cyclic codes of length $n$ and the factors of $X^{n}-1$ in $\mathbb{F}_{2}[X]$.

(b) What is a linear feedback shift register?

Suppose that $M: \mathbb{F}_{2}^{d} \rightarrow \mathbb{F}_{2}^{d}$ is a linear feedback shift register. Further suppose $\mathbf{0} \neq \mathbf{x} \in \mathbb{F}_{2}^{d}$ and $k$ is a positive integer such that $M^{k} \mathbf{x}=\mathbf{x}$. Let $H$ be the $d \times k$ matrix $\left(\mathbf{x}, M \mathbf{x}, \ldots, M^{k-1} \mathbf{x}\right)$. Considering $H$ as a parity check matrix of a code $C$, show that $C$ is a binary cyclic code.

(c) Suppose that $C$ is a binary cyclic code. Prove that, if $C$ does not contain the codeword $11 . . .1$, then all codewords in $C$ have even weight.