Q$2$: BCH code that operates over the Galois field GF $(2^m)$ which has primitive polynomial $p(x)=x^3+x+1.$ Determine:

$(33$ M$)$

a$)$ The generator polynomial.

b$)C_{BCH}(n,k)$ and r $.$

c$)$ The parity check matrix in binary form.

d$)$ Encode the message $0111.$

e$)$ Decode the received message if an error occurs in the $6^{th}$ bit.