Need Linear Programming expert. No AI solution

Question $1.$ In class we discussed the famous Hamming code. This code is an instance

of what is called a linear code, because the code itself is a linear subspace of $\{0,1{\}}^7.$ The

$(7,4)$ Hamming code communication scheme encodes a vector xin$\{0,1{\}}^4$ by $Hx$ where

$H=([1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[0,1,1,1],[1,0,1,1],[1,1,0,1]).$

$($a$)$ Show that the minimum distance between any two distinct vectors $Hx,H{x}^{\text{'}}$ in the Ham$-$

ming code is at least $3.$

$($b$)$ Show that the size of the Hamming code is optimal, i$.$e$.,$ there is no code Csub$\{0,1{\}}^7$

with $|C|17$ and minimum distance $3.$