Consider a $(n,k)$ systematic linear block code whose parity$-$check equations are written by

$p_1=m_1+m_3+m_4$

$p_2=m_2+m_3+m_4$

$p_3=m_1+m_2+m_3$

where $m_i,i=1,2\mathrm{.}3,4,$ are message bits and $p_i,i=1,2,3,$ are parity bits.

$($a$)$ Find the values of $n,k$ and $d_{\text{m}\text{i}\text{n}},$ where $d_{\text{m}\text{i}\text{n}}$ is the minimum distance.

$($b$)$ How many errors can this code correct?

$($c$)$ Find the parity$-$check matrix $H$ in the form of $P^T$;$I_{n-k}.$