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

A recent result of Zheng and Tse states that over a quasi-static channel, there exists a fundamental tradeoff, referred to as the diversity–multiplexing gain (D-MG) tradeoff, between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space–time (ST) code. This tradeoff is precisely known in the case of independent and identically distributed (i.i.d.) Rayleigh fading, for
$T geq n_t+n_r-1$
where
$T$
is the number of time slots over which coding takes place and
$n_t,n_r$
are the number of transmit and receive antennas, respectively. For
$T ≪ n_t+n_r-1$
, 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
$T=n_t=n$
, 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
$n$
, we provide here a construction for such codes that is valid for all
$n$
. For the rectangular,
$T ≫ n_t$
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
$Tgeq n_t$
is the same as that previously known to hold for
$T geq n_t + n_r -1$
.