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Abstract-A new systematic diagonal space-time block code (cyclotomic space-time code) design is proposed by using high dimensional cyclotomic lattices and some algebraic number theory. This design can be applied to any number of transmit antennas and for a fixed number of transmit antennas, there are infinitely many cyclotomic space-time codes. It is shown that all the cyclotomic space-time codes from the design have full diversity. Furthermore, optimal cyclotomic space-time codes from the design for different numbers of transmit antennas are obtained, where the optimality is in the sense that, for a fixed mean transmission signal power, its diversity product is maximized, or for a fixed diversity product, its mean transmission signal power is minimized. Although there are some cyclotomic space-time codes existed in the literature, most of them are not optimal.