Counting hypergraph colorings in the local lemma regime

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≥ 28 and q >357Delta^frac14k-14 . We also obtain a polynomial-time almost uniform sampler if q>931Delta^frac16k-16/3. These are the first approximate counting and sampling algorithms in the regime qllDelta (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.