Exercise Nr$.6(20$ points$)$ Gray codes have a useful property in that consecutive numbers differ in only a single bit position. The following table lists a $4-$bit Gray code representing the numbers $0$ to $15$:

$\backslash$table$[[$Number$,$Gray code$],[0,0000],[1,001],[2,\backslash$table$[[0,0,1,1]]],[3,0,010$