Explicit space-time codes that achieve the diversity-multiplexing gain tradeoff

In the recent landmark paper of Zheng and Tse it is shown for the quasi-static, Rayleigh-fading MIMO channel with n_t transmit and n_r receive antennas, that there exists a fundamental tradeoff between diversity gain and multiplexing gain, referred to as the diversity-multiplexing gain (D-MG) tradeoff. This paper presents the first explicit construction of space-time (ST) codes for an arbitrary number of transmit and/or receive antennas that achieve the D-MG tradeoff. It is shown here that ST codes constructed from cyclic-division-algebras (CDA) and satisfying a certain non-vanishing determinant (NVD) property, are optimal under the D-MG tradeoff for any n_t,n_r. Furthermore, this optimality is achieved with minimum possible value of the delay or block-length parameter T = n _t. CDA-based ST codes with NVD have previously been constructed for restricted values of n_t. A unified construction of D-MG optimal CDA-based ST codes with NVD is given here, for any number n_t of transmit antennas. The CDA-based constructions are also extended to provide D-MG optimal codes for all T ges n_t, again for any number nt of transmit antennas. This extension thus presents rectangular D-MG optimal space-time codes that achieve the D-MG tradeoff. Taken together, the above constructions also extend the region of T for which the D-MG tradeoff is precisely known from T ges n_t + n_r - 1 to T ges n_t