  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,
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 $q\bar{q}$-expansion of the partition
function in the twisted sectors of asymmetric orbifolds.
