Geometry optimizations are performed for three polytypes of h-BN using density functional theory with dispersion correction for the van der Walls interaction. Quasiparticle band structure calculations are carried out to solve the controversy on band gap type of h-BN. Band energies are corrected by GW method. The h-BN with Bk structure has an indirect band gap of 5.840eV. Two kinds of h-BN polytypes are shown to be mechanically stable and have quasi-direct band gap type. ► Structures of hexagonal boron nitride polytypes are studied by dispersion-corrected DFT. ► Band gap characters of hexagonal boron nitride are predicted by GW method. ► Hexagonal boron nitride polytype with quasi-direct band gap type is probably exists. ► GW band-gap energies are used to interpret recently reported experimental results.