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In this paper, we reformulate some constructions of real and complex diagonal lattice space-time codes from number fields which have been given explicitly or implicitly by other researchers. These constructions establish a connection between good diagonal lattice space-time codes and number fields with small absolute values of discriminants. We present two tables for diversity products of some lattice space-time codes from these constructions. The maximal rank of diagonal lattice space-time codes with positive diversity product is determined. We also discuss the asymptotic problem of lattice space-time codes. By using an infinite tower of Hilbert class field and a tamely ramified class field tower, we obtain asymptotically good sequences of lattice space-time codes. Some asymptotic upper bounds are given in the paper as well.