The Perfect Space Time Block Codes

Definition

Perfect Codes are Space-Time codes for the coherent MIMO channel.
They were defined in [ORBV06]. They are algebraic codes, built on non-commutative fields (or division algebras ).
The channel model considered is the following: if M is the number of transmit and receive antennas,

Y = H X + N (1)

where H ={hij} is the MxM channel matrix with complex fading coefficients and N the MxM complex Gaussian noise matrix.

A perfect space-time code satisfies by definition the following properties:

Full-rank : the determinant of the difference of any two distinct codewords is different from 0.
Full-rate : all the M² degrees of freedom of the system are used, which allows to send M² information symbols, either QAM or HEX.
Non-vanishing determinant for increasing rate : the minimum determinant of a perfect code is lower bounded away from zero by a constant. This constant, prior to SNR normalization, does not depend on the spectral efficiency.
Efficient shaping : The energy required to send the linear combination of the information symbols on each layer is similar to the energy used for sending the symbols themselves. This can be interpreted by saying that each layer is carved from a rotated version of the lattice Z[i]^M or A₂^M, where A₂ is the hexagonal lattice.
It achieves the Diversity Multiplexing Gain Trade-off [ERPVL06].
Uniform energy : It induces uniform average transmitted energy per antenna in all T=M time slots, i.e., all the coded symbols in the code matrix have the same average energy.

Code Construction

Perfect codes only exist in dimension 2, 3, 4, and 6 [BO06]. Codewords of a Perfect code have the form:

SUM [diag(Mu_j) E^j-1, j=1...M] (2)

where u_{j =}[u_j,1, .... , u_j,M] , M is a MxM unitary matrix defined below, and

	0	1	...		0
	0	0	1
E =			...	1
	0			...	1
	g	0	...	0	0

For M=2 antennas, QAM symbols are sent. There are infinitely many of them, but the most famous is the Golden Code
For M=3 antennas, HEX symbols are sent. (g = exp(2πi/3) and M). The minimum determinant is 1/49.
For M=4 antennas, QAM symbols are sent. (g = i and M). The minimum determinant is 1/1125.
For M=6 antennas, HEX symbols are sent. (g = -exp(2πi/3) and M). The minimum determinant is between 1/(2⁶ 7⁴) and 1/(2⁶ 7⁵) .

Decoder

The Sphere Decoder can be applied to decode the Perfect codes by vectorizing (1) and using (2) similarly to the Golden code.

References

[ORBV06]	F. Oggier, G. Rekaya, J.-C. Belfiore, E. Viterbo: "Perfect Space-Time Blocks Codes," IEEE Transactions on Information Theory, Sep. 2006.
[ERPVL06]	P. Elia, K. Raj Kumar, S. A. Pawar, P. Vijay Kumar and H.-F. Lu :"Explicit, Minimum-Delay Space-Time Codes Achieving the Diversity-Multiplexing Gain Tradeoff," IEEE Transactioins on Information Theory , to appear, 2006.
[BO06]	G. Berhuy, F. Oggier: "On the Existence of Perfect Codes," submitted to IEEE Transactioins on Information Theory, 2006.