Define the binary Hamming code of length $n=2^{\ell}-1$ and its dual. Prove that the Hamming code is perfect. Prove that in the dual code:

(i) The weight of any non-zero codeword equals $2^{\ell-1}$;

(ii) The distance between any pair of words equals $2^{\ell-1}$.

[You may quote results from the course provided that they are carefully stated.]