(i) Describe the original Hamming code of length 7 . Show how to encode a message word, and how to decode a received word involving at most one error. Explain why the procedure works.

(ii) What is a linear binary code? What is its dual code? What is a cyclic binary code? Explain how cyclic binary codes of length $n$ correspond to polynomials in $\mathbb{F}_{2}[X]$ dividing $X^{n}+1$. Show that the dual of a cyclic code of length $n$ is cyclic of length $n$.

Using the factorization

$$X^{7}+1=(X+1)\left(X^{3}+X+1\right)\left(X^{3}+X^{2}+1\right)$$

in $\mathbb{F}_{2}[X]$, find all cyclic binary codes of length 7 . Identify those which are Hamming codes and their duals. Justify your answer.