$7\mathrm{.}31$ Let $C$ be the binary cyclic code of length $15$ generated by $g(x)=1+x^2+x^4+x^5.$

$($i$)$ Find the minimum distance of $C.$

$($ii$)$ Show that $C$ can correct all bursts of length $2$ or less.

$($iii$)$ Decode the following received words using burst$-$error$-$correction:

$($a$)010110000000010$ ; $($b$)110000111010011.$