Finding generator polynomials of binary cyclic codes when gcd(blocklength,2) is greater than 1?

Question

I have just stumbled across the exercise "Find the generator polynomial and the minimum distance for each binary cyclic code of blocklength 8" and am now wondering how to solve it. If the block length would not share a common divisior greater 1 with 2, I know I could factor $x^8$-1 with the cyclotomic cosets.. but here I dont even know where to start. Is $x^8$ -1 already the generator polynomial? Thanks

Answer 1

Over $GF(2)$, we have $x^8-1=(x-1)^8$. So the divisors of $x^8-1$ are the products $(x-1)^k$, with $0\leq k\leq 8$.