Normal bases via general Gauss periods

Gauss periods have been used successfully as a tool for constructing normal bases in finite fields. Starting from a primitive $r$th root of unity, one obtains under certain conditions a normal basis for ${\mathbb F}_{q^n}$ over ${\mathbb F}_q$, where $r$ is a prime and $nk=r-1$ for some integer $k$. We generalize this construction by allowing arbitrary integers $r$ with $nk=\varphi(r)$, and find in many cases smaller values of $k$ than is possible with the previously known approach.