$($a$)$ Prove: If $(n,k,3)$ satisfies the Hamming bound, then there exists a binary linear $(n,k,3)$ code.

$($b$)$ Show that the above is not true for $(n,k,4).$ That is$,$ there exist $n$ and $k$ such that $(n,k,4)$ satisfies the Hamming bound, but there is no binary linear $(n,k,4)$ code.