Let $C$ be the Hamming $(n, n-d)$ code of weight 3 , where $n=2^{d}-1, d>1$. Let $H$ be the parity-check matrix of $C$. Let $\nu(j)$ be the number of codewords of weight $j$ in $C$.

(i) Show that for any two columns $h_{1}$ and $h_{2}$ of $H$ there exists a unique third column $h_{3}$ such that $h_{3}=h_{2}+h_{1}$. Deduce that $\nu(3)=n(n-1) / 6$.

(ii) Show that $C$ contains a codeword of weight $n$.

(iii) Find formulae for $\nu(n-1), \nu(n-2)$ and $\nu(n-3)$. Justify your answer in each case.