Transversals and diagonals of iterated groups and quasigroups of large arity

For the large-dimensional Latin hypercubes which are Cayley tables of iterated quasigroups the asymptotic behavior of the number of transversals is refined. In the case of iterated groups, the exact value of this asymptotics was found depending on the fulfillment of the Hall–Paige condition. In addition, estimates on the number of near transversals in iterated quasigroups of large arity were obtained.