Asymmetric Z₄ orbifolds of type IIB string theory revisited

We construct freely acting asymmetric Z₄ orbifolds of type IIB string theory on T⁵ preserving 24,16 or 8 supercharges in five dimensions. We show that these models are well-defined if the SO(8) lattice is chosen, while the SU(2)⁴ lattice, which was previously considered in the literature, does not lead to a modular invariant partition function. Also, we clarify some issues related to the coefficients in the qq̄-expansion of the partition function in the twisted sectors of asymmetric orbifolds.