$7\mathrm{.}25$ Find the generating idempotent for each of the following cyclic codes:

$($a$)$ the binary $7,4-$Hamming code Ham$(3,2)$;

$($b$)$ the binary $15,11,3-$Hamming code Ham$(4,2)$;

$($c$)$ the ternary $13,10,3-$Hamming code Ham$(3,3).)($