The Golden Code

Definition

The Golden Code is a Space-Time code for 2 transmit and 2 receive antennas, for the coherent MIMO channel.
It has been found independently by [BRV05],[YW03],[DV03].
The channel model considered is Y = H X + N, where H ={hij} is the 2x2 channel matrix with complex fading coefficients
and N the 2x2 complex Gaussian noise matrix.
The codewords X of the Golden Code are 2x2 complex matrices of the following form :

X = 1/sqrt(5) *	α [a+bθ]	α [c+dθ]
X = 1/sqrt(5) *	i σ(α) [c+dσ(θ)]	σ(α) [a+bσ(θ)]

where

a,b,c,d are the information symbols which can be taken from any M-QAM constellation carved from Z[i]
i = sqrt(-1)
θ = (1+sqrt(5))/2 = 1.618... (Golden number)
σ(θ) = (1-sqrt(5))/2 = 1-θ
α = 1 + i - i θ = 1 + i σ(θ)
σ(α) = 1 + i - i σ(θ) = 1 + i θ

Using the relations θ σ(θ) = -1 and θ + σ(θ) = 1 we can rewrite the codeword matrices as:

X = 1/sqrt(5) *	[1 + i σ(θ)]a + [θ-i]b	[1 + i σ(θ)]c + [θ-i]d
X = 1/sqrt(5) *	[i - θ]c + [1 + iσ(θ)]d	[1 + i θ]a + [σ(θ)-i]b

Properties

Full-rank : the determinant of the difference of 2 codewords is always different from 0.
Full-rate : the four degrees of freedom of the system are used, which allows to send 4 information symbols.
Non-vanishing determinant for increasing rate : the minimum determinant of the Golden Code is 1/5.
Cubic shaping : each layer is carved from a rotated version of Z[i]^2.
It achieves the Diversity Multiplexing Frontier [YW03].
The spectral efficiency is 2log2(M) bits/s/Hz.

ML Decoding with the SphereDecoder

In order to decode the Golden Code, the matrix has to be vectorized, furthermore real and imaginary part are separated,
so as to obtain an 8x8 matrix R, as shown below.

R = 1/sqrt(5) *

1	-σ(θ)
σ(θ)	1

θ	1
-1	θ

0	0
0	0

0	0
0	0

0	0
0	0

0	0
0	0

-θ	-1
1	-θ

1	-σ(θ)
σ(θ)	1

0	0
0	0

0	0
0	0

1	-σ(θ)
σ(θ)	1

θ	1
-1	θ

1	-θ
θ	1

σ(θ)	1
-1	σ(θ)

0	0
0	0

0	0
0	0

The Golden Code can be seen as a rotated Z⁸ algebraic lattice [OV04], with an orthogonal generator matrix R, and sent over a channel described by an 8x8 matrix H'

H' =

Re(h11)	-Im(h11)
Im(h11)	Re(h11)

Re(h12)	-Im(h12)
Im(h12)	Re(h12)

0	0
0	0

0	0
0	0

Re(h21)	-Im(h21)
Im(h21)	Re(h21)

Re(h22)	-Im(h22)
Im(h22)	Re(h22)

0	0
0	0

0	0
0	0

0	0
0	0

0	0
0	0

Re(h11)	-Im(h11)
Im(h11)	Re(h11)

Re(h12)	-Im(h12)
Im(h12)	Re(h12)

0	0
0	0

0	0
0	0

Re(h21)	-Im(h21)
Im(h21)	Re(h21)

Re(h22)	-Im(h22)
Im(h22)	Re(h22)

The received vector becomes y' = H'Rx'+ n, where n is the real Gaussian noise vector and x'

x' =	Re(a)
	Im(a)
	Re(b)
	Im(b)
	Re(c)
	Im(c)
	Re(d)
	Im(d)

The decoding of the 8-dimensional lattice with generator matrix M=H'R can be performed using the Sphere Decoder [VB98].

Performance of the Golden Code (*)

Codeword Error Rate

Bit Error Rate

(*) We thank Barbara Cerato for generating these simulation curves.
Performance in real world scenarios can be found in [MRRB08].

References

[BRV05]	J.-C. Belfiore, G. Rekaya, E. Viterbo: "The Golden Code: A 2 x 2 Full-Rate Space-Time Code with Non-Vanishing Determinants," IEEE Transactions on Information Theory, vol. 51, n. 4, pp. 1432-1436, Apr. 2005.
[DV03]	P. Dayal, M.K. Varanasi: "An Optimal Two Transmit Antenna Space-Time Code and its Stacked Extensions," Proceedings of Asilomar Conf. on Signals, Systems and Computers, Monterey, CA , November 2003.
[ORBV05]	F. Oggier, G. Rekaya, J.-C. Belfiore, E. Viterbo: "Perfect Space Time Block Codes," submitted to IEEE Transactions on Information Theory, Sep. 2004.
[OV04]	F. Oggier and E. Viterbo, "Algebraic number theory and code design for Rayleigh fading channels," Foundations and Trends in Communications and Information Theory, vol. 1, pp. 333-415, 2004.
[RBV05]	G. Rekaya, J.-C. Belfiore, E. Viterbo: "Algebraic 3x3, 4x4 and 6x6 Space-Time Codes with Non-Vanishing Determinants," Proceedings of Intern. Symp. on Inform. Theory and its Applications, ISITA , October 2004, pp. 325-329.
[VB98]	E. Viterbo and J. Boutros: "A Universal Lattice Code Decoder for Fading Channels," IEEE Transactions on Information Theory, vol. 45, n. 5, pp. 1639-1642, July 1999.
[YW03]	H. Yao, G.W. Wornell: "Achieving the Full MIMO Diversity-Multiplexing Frontier with Rotation-Based Space-Time Codes," Proceedings of Allerton Conf. on Communication, Control and Computing , October 2003.
[MRRB08]	Lina Mroueh, Stephanie Rouquette-Leveil, Ghaya Rekaya-Ben Othman, and Jean-Claude Belfiore: "On the performance of the Golden code in BICM-MIMO system and in IEEE 802.11n standard," Forty-First Asilomar Conference on Signals, Systems and Computers, 2007. Nov, 2007.