What's in general de minimum distance of cyclic codes of length 2^m?

Question

I've been asked to first verify that $(1+x)^{2^m} = 1+x^{2^m}$ in $\mathbb{F}_2 [x]$, then to make a prediction of the minimum distance of cyclic codes of length and proof such prediction. We managed to figure out that with m=2 the minimum distance is 2 but I don't know if this could be a good prediction or how to prove that all codes of lenght $2^m$ is 2.