Design maximizing the ergodic channel capacity: linear dispersion (LD) codes  are designed for multiple antenna channels by a search that maximizes the ergodic capacity under a gaussian channel input. Such a design is not necessarily suitable for a block fading channel with a finite number of states,
, etc. Also, the kind of input alphabet is not considered in the search for linear dispersion codes.
Design based on genie conditions: this construction was introduced in 
where perfect a priori probability feedback is assumed in the iterative joint detection
and decoding of ST-BICM. In order to guarantee maximum diversity order and maximum coding gain
at the output of the detector, the design must guarantee two conditions:
1- Orthogonal sub-rows in the linear precoding matrix,
and 2- Equal norm sub-rows in the linear precoding matrix.
In ST-BICM, there exists a strong interaction between the error correcting code with interleaving
and the linear precoder, both in terms of diversity and coding gain maximization .
Complexity can be controlled by the choice of a minimal spreading factor that guarantees full diversity .
The genie conditions are optimal, in terms of ML performance,
when all diversity given by the transmit antennas is collected at the detector (i.e. ).
A supplementary condition called ``Dispersive Nucleo Algebraic'' (DNA) has been proposed in
 to keep optimality when
while having the genie conditions on sub-groups of transmit antennas.
As an example, the cyclotomic rotation given below is an algebraic precoder satisfying the genie conditions for ST-BICM with :
Consider the space-time bit-interleaved coded modulation drawn in Fig. 1. This ST-BICM is a serial concatenation of a rate binary convolutional code, an interleaver of size bits, and a QAM mapper followed by the precoder as described in the previous section. When is the identity matrix, the ST-BICM diversity order is upper-bounded by :
With a vanishing coding rate, i.e. , it is possible to attain the overall system diversity order produced by the receive antennas, the transmit antennas and the distinct channel states. Unfortunately, this is unacceptable due to the vanishing transmitted information rate. Precoding is one means to achieve maximum diversity with a non-vanishing coding rate. Under linear precoding that spreads QAM symbols over time periods, the Singleton bound becomes :
Now if , from the above inequality, we observe that precoding may achieve maximal diversity
without the use of error-correcting codes. Unfortunately, near outage performance is impossible
in this case due to the weak coding gain of all kinds of precoders. The near-outage performance
of ST-BICM is a judicious trade-off between error-control coding and linear QAM precoding.
Hence, we propose a simple information theoretical design of multi-dimensional rotations
that take into account the interaction between channel coding and symbol space-time spreading.
Joseph Jean Boutros 2005-05-07