  We give a fully polynomial-time approximation scheme (FPTAS) to count the
number of $q$-colorings for $k$-uniform hypergraphs with maximum degree
$\Delta$ if $k\ge 28$ and $q >357\Delta^{\frac{14}{k-14}}$ . We also obtain a
polynomial-time almost uniform sampler if $q>931\Delta^{\frac{16}{k-16/3}}$.
These are the first approximate counting and sampling algorithms in the regime
$q\ll\Delta$ (for large $\Delta$ and $k$) without any additional assumptions.
Our method is based on the recent work of Moitra (STOC, 2017). One important
contribution of ours is to remove the dependency of $k$ and $\Delta$ in
Moitra's approach.
