# Space-Time coding uses large amount of block codes to acheive maximum spectral efficiency

Assume I have a MIMO system with $$M=8$$ transmit antennas, and an arbitrary number of receive antennas.

This MIMO systems uses space-time coding. the block codes (matrices) transmitted are part of a set $$S$$. $$S$$ contains 110592 unitary matrices having complex elements belonging to an M-PSK moduolation.

And let $$T$$ be the normalized coherence interval, which is the duration in which the channel remains constant.

The maximum spectral efficiency achievable by such a system is: $$R_{max}=\frac{1}{T}\left\lfloor \log_2 |S| \right\rfloor$$ where $$\left\lfloor \log_2 |S| \right\rfloor$$ is the cardinality of the set $$S$$, which is 110592. This leads to a maximum spectral efficiency of $$2\text{ bps/Hz}$$.

The number of matrices needed to attain $$2\text{ bps/Hz}$$ or $$\text{bpcu}$$ (bits per channel use) is $$2^{RT}$$ which amounts to a set $$C\subset S$$ of 65536 matrix.

In my work, these 65536 matrix are not chosen randomly from the set $$S$$, rather they follow a criterion which is to choose the matrices having the largest distance among each other in order to improve system performance. The distance between two matrices is measured using the Frobenius Norm as follows $$D_{MaMb}=||M_a-M_b||_F$$

My problem is that it is not feasible to run my search algorithm on all 65536 matrices. Is there a sub-optimal method that i could use to select these matrices without running through the whole Set?

• T is 8 as you said. And indeed i am using differential spatial modulatiom. – Ibrahim DAWI Nov 14 '19 at 13:32
• What you're describing is the very fundamental problem of all block code design! Finding the set of codewords $C$ from a larger set of equally long sequences. – Marcus Müller Nov 14 '19 at 13:33
• So, what's the mathematical structure of $S$? Is it $S\subset \mathbb C^{M\times N}$? – Marcus Müller Nov 14 '19 at 13:33
• $S$ contains unitary matrices having complex elements belonging to an M-PSK modulation. – Ibrahim DAWI Nov 14 '19 at 13:34
• Nice, could you add that to the question by editing it? Makes it easier for future readers ! Also, add the specific metric you use to define "distance", that maps two matrices to their distance, so that people have something to optimized :) – Marcus Müller Nov 14 '19 at 13:45