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A recent result of Zheng and Tse states that over a quasi-static channel, there exists a fundamental tradeoff, referred to as the diversity&#8211;multiplexing gain (D-MG) tradeoff, between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space&#8211;time (ST) code. This tradeoff is precisely known in the case of independent and identically distributed (i.i.d.) Rayleigh fading, for<tex>$T geq n_t+n_r-1$</tex>where<tex>$T$</tex>is the number of time slots over which coding takes place and<tex>$n_t,n_r$</tex>are the number of transmit and receive antennas, respectively. For<tex>$T &#8810; n_t+n_r-1$</tex>, only upper and lower bounds on the D-MG tradeoff are available. In this paper, we present a complete solution to the problem of explicitly constructing D-MG optimal ST codes, i.e., codes that achieve the D-MG tradeoff for any number of receive antennas. We do this by showing that for the square minimum-delay case when<tex>$T=n_t=n$</tex>, cyclic-division-algebra (CDA)-based ST codes having the nonvanishing determinant property are D-MG optimal. While constructions of such codes were previously known for restricted values of<tex>$n$</tex>, we provide here a construction for such codes that is valid for all<tex>$n$</tex>. For the rectangular,<tex>$T &#8811; n_t$</tex>case, we present two general techniques for building D-MG-optimal rectangular ST codes from their square counterparts. A byproduct of our results establishes that the D-MG tradeoff for all<tex>$Tgeq n_t$</tex>is the same as that previously known to hold for<tex>$T geq n_t + n_r -1$</tex>.

Explicit Space&#8211;Time Codes Achieving the Diversity&#8211;Multiplexing Gain Tradeoff

Explicit Space–Time Codes Achieving the Diversity–Multiplexing Gain Tradeoff