Let $K$ be an algebraic closure of $\mathbb{F}_2$. The cyclic group $C_{2^n}$ acts on $K[x_0, \dots, x_{2^n-1}]$ by cyclically permuting the $x_i$: $a : x_i \rightarrow x_{i + a \bmod 2^n}$. Is there a nice description of the ring of invariants of $C_{2^n}$ acting this way on $K[x]$? Things are quite easy when the characteristic $ \ne 2$, but look quite a bit more intricate here since, e.g. the group ring $K[C_{2^n}]$ is not semi-simple.