$($A$)$ Consider the primitive polynomial $p(x)=x^4+x+1,$ design a single$-$error$-$correcting systematic BCH code to decode the received word $r(x)=x^6+x^4+x^3+x^2+1,$ using Peterson's method.

$($B$)$ Suggest a digital circuit to implement the encoder in $($A$).$