Asymmetric ${\mathbb{Z}}_4$ orbifolds of type IIB string theory revisited

We construct freely acting asymmetric ${\mathbb{Z}}_4$ orbifolds of type IIB string theory on $T^5$ preserving 24,16 or 8 supercharges in five dimensions. We show that these models are well-defined if the SO(8) lattice is chosen, instead of the SU(2)${}^4$ lattice, which was previously considered in the literature.