The Silver Code



The Silver Code is a fast-decodable space-time 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 re-found by [PGA07][SF07], which pointed out its fast-decoding properties and it was also summarized by [BHV09].

The channel model considered is Y = H X + N, where H ={h
ij} 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 = Xa(s1,s2) + TXb(z1,z2)


  • Xa and Xb take Alamouti structure

Xa(s1,s2) =




Xb(z1,z2) =




[z1, z2]T = U*







si , i=1,...,4, are the information QAM symbols

  • U is an unitary matrix, and the optimum U matrix is given by the following in order to maximize the minimum determiant of the codeword matrix X

U = 1/sqrt(7) *

1 + j

-1 + 2j

1 + 2j

1 - j

  • T is chosen as the following matrix to gaurantee the cubic shaping property [BHV09]


              T =






  1. Full-rank : the determinant of the difference of 2 codewords is always different from 0.
  2. Full-rate : the four degrees of freedom of the system are used, which allows to send 4 information symbols.
  3. Non-vanishing determinant for increasing rate [HLRVV08]: the minimum determinant is 1/7,slightly smaller than that of the Golden code.
  4. Cubic shaping : each layer is carved from a rotated version of Z[i]^2.
  5. The spectral efficiency is  2log2(Q) bits/s/Hz for Q-QAM.
  6. It achieves the Diversity Multiplexing Frontier.
  7. Fast-decoding property: the worst-case maximum likelihood decoding (MLD) complexity is 2M3, as compared to a standard MLD complexity M4, where M is the cardinality of the signal constellation.

Reduced-Complexity MLD with the SphereDecoder

         Consider a linear space-time 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 = {si}, 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 = [f1 | f2 | f3 | f4], where fi 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 =




<f3, e1>

<f4, e1>


|| d2||

<f3, e2>

<f4, e2>



|| d3||





|| d4||


where di = fi – sum( Projej fi, j = 1, …, i-1), where Projuv = <v,u>/<u,u> and ei = di/||di||.

         Note that there are two zeros in the matrix R which lead to a reduced-complexity MLD [PGA07][SF07][BHV09].

1) <f2, e1> = 0 provides a saving of 2-dimensional complex SD tree search, i.e., we employ 2-dimensional complex SD tree search to find s3, s4, with complexity of M2. For the remaining pair (s1,s2), an Alamouti decoding is used with decoding complexity 2M. In summary, the worst-case decoding complexity is 2M3.

2)  <f4, e3> = 0 leads to a faster metric computation in the relevant SD computation.


Performance of the Silver Code, compared to Golden Code (*)





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 IEEE Trans.
On Information Theory, pp. 524-530, vol. 55, n. 2, Feb. 2009.


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Last modified 29/5/2008 by Yi Hong