(i) Answer the following questions briefly but clearly.

(a) How does coding theory apply when the error rate $p>1 / 2$ ?

(b) Give an example of a code which is not a linear code.

(c) Give an example of a linear code which is not a cyclic code.

(d) Give an example of a general feedback register with output $k_{j}$, and initial fill $\left(k_{0}, k_{1}, \ldots, k_{N}\right)$, such that

$$\left(k_{n}, k_{n+1}, \ldots, k_{n+N}\right) \neq\left(k_{0}, k_{1}, \ldots, k_{N}\right)$$

for all $n \geqslant 1$.

(e) Explain why the original Hamming code can not always correct two errors.

(ii) Describe the Rabin-Williams scheme for coding a message $x$ as $x^{2}$ modulo a certain $N$. Show that, if $N$ is chosen appropriately, breaking this code is equivalent to factorising the product of two primes.