I have to write down a quasi parity check matrix of a BCH code of lenth 63 over $\mathbb{F}_{2^2}=\{0, 1, a, b=1+a\}$, then decode the vector $$a11abb10babbb01b1baaabba11bbaabaaaa001b0b1a1aa110110110b11b10bb \in \mathbb{F}_{2^2}^{63}.$$
I can decode vectors when the base field is $\mathbb{F_2}$, but I'm having difficulties doing the same here. We can't use computers for this(!), the problem is from a previous written exam.
Edit: it might be of use to know $p(x)=x^3 + x^2+ax+b$ is a primitive irreducible polynomial.