Algebraic number theory provides effective means to construct rotated Zn
lattices with full diversity and large minimum product distance. These
two properties enable to design good signal constellations for the
independent Rayleigh fading channel. The following tables provide the
best known constructions for these lattices in terms of highest minimum
product distance. The corresponding rotated Zn
lattice generator matrices are orthogonal matrices (i.e., M*Mt=In) and can be downloaded in text
form.
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3 | exp(2πi/13) |
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1 |
1/13 | 0.425290 | 0.52461 |
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r=2, lambda=1 |
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sqrt(5) |
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r=2, lambda=1 |
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0.275946 |
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2cos(2π/7) |
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+10x3-4x2-4x+1 |
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8 | 2cos(2π/32) | 1/2(31/2) | 0.261068 |
0.30093 | dast_8 | [DAB02] | ||
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2cos(2π/5)
2cos(2π/11)
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cyclo_2Xcyclo_5 | ||
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r=5, lambda=6 |
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2cos(2π/5)
2cos(2π/13)
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mixed_2x6 |
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12 | 1/sqrt(494*56*293) |
0.229487 |
0.24807 |
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cyclo_3Xkrus_4 | |||
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16 | 2cos(2π/64) | 1/2(79/2) | 0.180648 |
0.21965 | dast_16 | [DAB02] | ||
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cyclo_3Xcyclo_6 | |||
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21 |
1/sqrt(497*201343933) |
0.157250 |
0.19803 | mixed_3x7 |
krus_3Xkrus_7 |
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mixed_2x11 |
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r=5, lambda=12 |
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krus_4Xkrus_7 |
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r=2, lambda=1 |
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cyclo_5Xcyclo_6 |
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Dimension |
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Primitive element of the algebraic number
field Q(θ): - for cyclotomic constructions K=Q(θ) - for cyclic constructions K is a subfield of Q(θ) |
I | Ideal of the ring of integers OK |
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Twisting element in the ideal lattice construction |
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Minimum product distance |
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Normalized minimum product distance (in red best known value, in boldface red optimal) |
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Upper bound on dp,min1/n based
on Odlyzko's bound for the root discriminant |
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Generator matrix in the row vector convention |
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Bibliographic reference |
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Ring of integers of K |
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Ideal of OK such that AK2 is the codifferent of K |
[O05] | F. Oggier, Algebraic methods for channel coding, Ph.D. Thesis, EPFL, 2005. |
[OV04] | F. Oggier and E. Viterbo, "Algebraic number theory and
code design for Rayleigh fading channels," in Foundations and Trends in Communications and Information Theory, vol. 1, pp. 333-415, 2004. |
[BOV04] | E. Bayer-Fluckiger, F. Oggier, E. Viterbo: "New
Algebraic Constructions of Rotated Z^n-Lattice Constellations for the
Rayleigh Fading Channel," IEEE Transactions on Information Theory, vol. 50, n. 4, pp. 702-714, Apr. 2004. |
[OB03] | F. Oggier, E. Bayer-Fluckiger, "Best rotated
cubic lattice constellations for the Rayleigh fading channel," Proceedings of the IEEE International Symposium on Information Theory, Yokohama, Japan, 2003. |
[DAB02] | M.O. Damen, K. Abed-Meriam, J.C. Belfiore:"Diagonal
algebraic space-time block codes," IEEE Transactions on Information Theory, vol. 48, pp. 628-636, Mar. 2002. |
[BV98] |
J. Boutros and E. Viterbo:
"Signal Space Diversity: a power and bandwidth efficient diversity
technique for the Rayleigh fading channel", IEEE Transactions on Information Theory, vol. 44, n. 4, pp. 1453-1467, July 1998. |
[GBB97] |
X.
Giraud, E. Boutillon, and J.C. Belfiore, "Algebraic tools to build modulation
schemes for fading channels" IEEE Transactions on Information Theory, vol. 43, n. 3, pp. 938 - 952, May 1997. |
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