An engineer has prepared a $(3,2)$ cyclic code in the $GF(3)$ domain such that $g(x)*h(x)=x^n+1.$ The parity checker polynomial is $a{x}^2+bx+1$ and the generator polynomial is

$(cx+1).$

$($i$)$ Identify the values of $a,b$ and $c.$

$($ii$)$ Determine the codeword for the message $21$ in the non$-$systematic form.d c$.$