A construction of a space-time code based on number theory

Abstract

We construct a full data rate space-time (ST) block code over M=2 transmit antennas and T=2 symbol periods, and we prove that it achieves a transmit diversity of 2 over all constellations carved from Z[i]/sup 4/. Further, we optimize the coding gain of the proposed code and then compare it to the Alamouti code. It is shown that the new code outperforms the Alamouti (see IEEE J Select. Areas Commun., vol.16, p.1451-58, 1998) code at low and high signal-to-noise ratio (SNR) when the number of receive antennas N>1. The performance improvement is further enhanced when N or the size of the constellation increases. We relate the problem of ST diversity gain to algebraic number theory, and the coding gain optimization to the theory of simultaneous Diophantine approximation in the geometry of numbers. We find that the coding gain optimization is equivalent to finding irrational numbers "the furthest," from any simultaneous rational approximations.