I am having (many) troubles with binary Goppa Codes. My question is at the moment: How do I calculate the trace on given points $tr(\alpha^u)$? For example: Given a finite field $GF(2^6)$ and the Goppa Polynomial
$G(z)= (z^2 + z + \alpha^{32}) * (z^2 + z + \alpha^{33}$)
and knowing that the cyclotomic cosets for $c_1$ and $c_3$ are
$c_1 = \{1,2,4,8,16,32\}$ and $c_2 = \{3,6,12,24,48,33\}$
how do I calculate $tr(\alpha^{32})$ and $tr(\alpha^{33})$?
How can I determine the code size $n = |L|$ and the parameters distance $d$ and the amount of information bits $k$?
UPDATE: The irreducible polynomial of the given finite field is $p(x) = x^6 + x^5 + x^3 + x^2 +1$. $\alpha$ is supposed to be the zero of $p(x)$ and is itself an element of the finite field, ideally a primitive element.