Construction of addition and multiplication table for GF(4)
This case is fairly easy because all the calculations are modulo 2 and the field has only 4 elements. Let $\alpha$ be a root of $g$, i.e, $\alpha^2 + \alpha + 1 = 0$. This immediately implies that $1+\alpha = -\alpha^2 = \alpha^2$, which you are calling $\beta$. I would suggest to try a polynomial of higher degree so that the field has more elements.