On optimal multilayer cyclotomic space-time code designs

High rate and large diversity product (or coding advantage, or coding gain, or determinant distance, or minimum product distance) are two of the most important criteria often used for good space-time code designs. In recent (linear) lattice-based space-time code designs, more attention is paid to the high rate criterion but less to the large diversity product criterion. In this paper, we consider these two criteria together for multilayer cyclotomic space-time code designs. In a previous paper, we recently proposed a systematic cyclotomic diagonal space-time code design over a general cyclotomic number ring that has infinitely many designs for a fixed number of transmit antennas, where diagonal codes correspond to single-layer codes in this paper. In this paper, we first propose a general multilayer cyclotomic space-time codes. We present a general optimality theorem for these infinitely many cyclotomic diagonal (or single-layer) space-time codes over general cyclotomic number rings for a general number of transmit antennas. We then present optimal multilayer (full-rate) cyclotomic space-time code designs for two and three transmit antennas. We also present an optimal two-layer cyclotomic space-time code design for three and four transmit antennas. The optimality here is in the sense that, for a fixed mean transmission signal power, its diversity product is maximized, or equivalently, for a fixed diversity product, its mean transmission signal power is minimized. It should be emphasized that all the optimal multilayer cyclotomic space-time codes presented in this paper have the nonvanishing determinant property.