The Silver Code is a fastdecodable spacetime
block code for 2 transmit and 2 receive antennas, for the coherent MIMO
channel. It has been found by [HT04] [HTWBook] [TH02] [TK02]. Recently it was
refound by [PGA07][SF07], which pointed out its fastdecoding properties and
it was also summarized by [BHV09].
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 = X_{a}(s_{1},s_{2})
+ TX_{b}(z_{1},z_{2}) 

where
X_{a}(s_{1},s_{2}) = 
s_{1} 
s_{2}^{*} 
X_{b}(z_{1},z_{2}) = 
z_{1} 
z_{2}^{*} 
[z_{1}, z_{2}]^{T} = U* 
s_{3} 
s_{2} 
s_{1}^{*} 
z_{2} 
z_{1}^{*} 
s_{4} 
s_{i} , i=1,...,4, are the information QAM
symbols
U = 1/sqrt(7) * 
1 + j 
1 + 2j 
1 + 2j 
1  j 
T = 
1 
0 
0 
1 
Porperties
• Consider a linear spacetime block
coded MIMO, where STBC carries 4 independent QAM information symboles. In a complex
vector form, we rephrase the received signal equation as vec(Y) = Fs
+ vec(N), where s = {s_{i}}, i=1,...,4, and
F = diag(H,...,H) x G, where G is the
generator matrix of the silver code given in [PGA07,BHV09], vec(·)
operator stacks the m column vectors of a n x m complex
matrix into a mn complex column vector.
• Let F = [f_{1}  f_{2}  f_{3}  f_{4}], where f_{i}
is a 4 dimensional column vector. Sphere decoding (SD) can be used to conduct
the MLD based on QR decomposition to minimize Q^{†} vec(Y)
– Rs, where (×)^{†} denotes matrix Hermitian, and F = QR, where
Q is a 4 x 4 unitary matrix, and R is a 4 x 4 upper triangular matrix
with the following special structure, where <a,b> denotes the inner product of a and b,
R =

d_{1} 
0 
<f_{3}, e_{1}> 
<f_{4}, e_{1}> 
0 
 d_{2} 
<f_{3}, e_{2}> 
<f_{4}, e_{2}> 

0 
0 
 d_{3} 
0 

0 
0 
0 
 d_{4} 
where d_{i} = f_{i} –
sum( Proj_{e}_{j }f_{i}, j
= 1, …, i1), where Proj_{u}v = <v,u>/<u,u> and e_{i} = d_{i}/d_{i}.
• Note that there are two zeros in the
matrix R which lead to a reducedcomplexity MLD [PGA07][SF07][BHV09].
1) <f_{2}, e_{1}> = 0 provides a saving of 2dimensional complex
SD tree search, i.e., we employ 2dimensional complex SD tree search to find s_{3},
s_{4}, with complexity of M^{2}. For the
remaining pair (s_{1},s_{2}), an Alamouti
decoding is used with decoding complexity 2M. In summary, the worstcase
decoding complexity is 2M^{3}.
2) <f_{4}, e_{3}> = 0 leads to a faster metric
computation in the relevant SD computation.
[HT04] 
A. Hottinen and O. Tirkkonen, ``Precoder designs for high rate spacetime block codes,'' in Proc. Conference on Information Sciences and Systems, Princeton, NJ, March 1719, 2004. 
[HTWBook] 
A.
Hottinen, O. Tirkkonen and R. Wichman, ``Multiantenna Transceiver Techniques
for 3G and Beyound,'' WILEY publisher, UK. 
[TH02] 
O. Tirkkonen and A. Hottinen, ``Squarematrix embeddable spacetime block codes for complex signal constellations,'' in IEEE Trans. Inform. Theory, vol. 48, no. 2, , pp. 384395, February 2002. 
[TK02] 
O.
Tirkkonen and R. Kashaev, ``Combined information and performance optimization
of linear MIMO modulations,'' in Proc IEEE Int. Symp. Inform. Theory
(ISIT 2002), 
[PGA07] 
J. Paredes,
A.B. Gershman, and M. G. Alkhanari, ``A2x2 spacetime code with
nonvanishing determinants and fast maximum likelihood decoding,'' in Proc
IEEE International Conference on Acoustics, Speech, and Signal Processing
(ICASSP2007), Honolulu, Hawaii, USA, pp. 877880, April 2007. 
[SF07] 
M. Samuel
and M. P. Fitz, ``Reducing the detection complexity by using 2x2 multistrata
spacetime codes,'' in Proc IEEE Int. Symp. Inform. Theory (ISIT
2007), pp. 19461950, Nice, 
[BHV09] 
E.
Biglieri, Y. Hong and E. Viterbo, "On fastdecodable spacetime block
codes," 
[HLRVV08] 
C. Hollanti,
J. Lahtonen, K. Ranto, R. Vehkalahti, and E. Viterbo, ``On the Algebraic
Structure of the Silver Code,'' in IEEE Information Theory Workshop,
Porto, Portugal, May 2008. 