SUBMITTED TO CISS 2005: SHORT PAPER 1 SPACE-TIME CODES MEETING THE DIVERSITY-MULTIPLEXING GAIN TRADEOFF WITH LOW SIGNALLING COMPLEXITY Hsiao-feng Lu, Petros Elia, K. Raj Kumar, Sameer A. Pawar, and P. Vijay Kumar Abstract In the recent landmark paper of Zheng and Tse it is shown that there exists a fundamental tradeoff between diversity gain and multiplexing gain, referred to as the Diversity-Multiplexing gain(D-MG) tradeoff. It is shown in [5] that ST codes with non-vanishing determinant (NVD) constructed from cyclic-division- algebras (CDA), are optimal under the D-MG tradeoff for any number n t ,n r of transmit and receive antennas and with minimum delay T = n t . CDA-based ST codes with NVD have been previously constructed for restricted values of n t . This paper presents an explicit construction of space-time (ST) codes for arbitrary number of transmit antennas that achieve the D-MG tradeoff. A unified construction of D-MG optimal CDA-based ST codes with NVD is given here, for any number n t of transmit antennas. Index Terms Space-time codes, diversity-multiplexing gain (D-MG) tradeoff, and algebraic number theory. I. I NTRODUCTION Consider the quasi-static, Rayleigh-fading space-time (ST) channel with quasi-static interval T , n t transmit, and n r receive antennas. The received signal matrix Y is given by Y = ρHX + W, (1) where X is the transmitted codeword, an (n t × T ) code matrix drawn from an ST code X , H the (n r × n t ) channel matrix and W the (n r × T ) noise matrix. The entries of H and W are assumed to be i.i.d., circularly symmetric, complex Gaussian CN (0, 1) random variables. The entries of X are drawn from a signal constellation whose size scales with SNR and ρ is chosen to ensure that E(‖ρX ‖ 2 F )= T SNR, (2) where ‖A‖ F denotes the Frobenius norm of matrix A. The ST code X transmits R = 1 T log(|X|) bits per channel use. For large SNR, Telatar [1] showed that the ergodic capacity of the ST channel model in (1) is given by C ≈ min{n t ,n r } log(SNR). In a recent landmark paper, Zheng and Tse [2] proved that there exists a fundamental tradeoff between diversity and multiplexing gain, referred to as the D-MG tradeoff. Let r be the normalized rate given by r = R/ log(SNR). Thus an ST code achieving normalized rate r has size |X | = SNR rT . It follows that the maximum achievable normalized rate equals r = min{n t ,n r }. Following [2], we will refer to r as the multiplexing gain. The diversity gain corresponding to a ST code X is defined by d(r)= − lim SNR→∞ log(P e ) log(SNR) , Hsiao-feng (Francis) Lu is with the Dept. of Comm. Engineering, National Chung-cheng University, 160 San-Hsing, Min- Hsiung, Chia-Yi 621, Taiwan, R.O.C. (francis@ccu.edu.tw). Petros Elia and P. Vijay Kumar are with the Department of EE- Systems, University of Southern California, Los Angeles, CA 90089 ({elia,vijayk}@usc.edu). K. Raj Kumar and Sameer A. Pawar are with the the Electrical Communication Engineering Department of the Indian Institute of Science Bangalore, 560 012 ({raj,sameerp}@ece.iisc.ernet.in). This work was carried out while P. Vijay Kumar was on sabbatical leave at the Indian Institute of Science Bangalore. This research is supported in part by NSF-ITR CCR-0326628, in part by the DRDO-IISc Program on Advanced Research in Mathematical Engineering and in part by NSC 93-2218-E-194-012.