Full diversity rotations

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.
The authors would be happy to hear about any contribution improving over the best known rotations reported here.


Table of best known full diversity algebraic rotations


 
n
 
θ
I
α
dp,min
dp,min1/n
Bound
Matrix
Notes&Refs
2
 (1+sqrt(5))/2
OK
3-θ 
1/sqrt(5)
 0.668740
0.66995
cyclo_2

2
 sqrt(2)
OK
 1/(2*sqrt(2)+4)
 1/(2*sqrt(2))
0.59460
0.66995
ideal_2
[BOV04]
3
2cos(2π/7)
OK
2-θ
1/7
0.522757
0.52461
cyclo_3
 as krus_3
3 exp(2πi/13)
AK
1
1/13 0.425290 0.52461
cyclic_3a
r=2, lambda=1
3
exp(2πi/9)
AK
1
1/9
0.480749
0.52461
cyclic_3b
r=2
4
x4-x3-3x2+x+1
 
 
1/sqrt(52*29)
0.438993
0.44163
krus_4
 
4
 sqrt(2)
sqrt(5)
 
 
1/40
0.397635
0.44163
mixed_2x2
cyclo_2Xideal_2
5
2cos(2π/11)
OK
2-θ
1/112
0.383215
0.38794
cyclo_5
as krus_5
5
exp(2πi/11)
AK
1
1/112
0.383215
0.38794
cyclic_5a
as cyclo_5
 r=2, lambda=1
5
exp(2πi/31)
AK
1
1/54
0.275946
0.38794
cyclic_5b
r=2
5
exp(2πi/25)
AK
1
1/312
0.253195
0.38794
cyclic_5c
r=3, lambda=16
6
2cos(2π/5)
2cos(2π/7)
 
 
1/sqrt(53*74)
0.349589
0.35032
mixed_2x3
as krus_6
6
2cos(2π/13)
OK
2-θ
1/13(5/2)
0.343444
0.35032
 cyclo_6
 
7
x7-x6-6x5+4x4
+10x3-4x2-4x+1 
 
 
1/sqrt(20134393)
0.300809
0.32245
krus_7
 
7
exp(2πi/29)
AK
1
1/293
0.236188
0.32245
cyclic_7a
r=2, lambda=1
7
exp(2πi/49)
AK
1
1/76 
0.188638
0.32245
cyclic_7b
r=3
8
2cos(2π/17)
OK
2-θ
1/17(7/2)
0.289520
0.30093
cyclo_8
 
8 2cos(2π/32)

1/2(31/2) 0.261068
0.30093 dast_8 [DAB02]
9
2cos(2π/19)
OK
2-θ
1/194
0.270187
0.28377
cyclo_9
 
10
2cos(2π/5)
2cos(2π/11)
   
1/sqrt(55*118)
0.256271
0.26973
 mixed_2x5
 cyclo_2Xcyclo_5
11
2cos(2π/23)
OK
2-θ
1/235
0.240454
0.25801
cyclo_11
 
11
exp(2πi/23)
AK
1
1/235
0.240454
0.25801
cyclic_11a
 as cyclo_11
r=5, lambda=6
11
exp(2πi/121)
AK
1
1/1110
0.113052
0.25801
cyclic_11b
 r=2
12
2cos(2π/5)
2cos(2π/13)
 
 
 1/(53*135)
0.229675
0.24807
mixed_2x6
cyclo_2Xcyclo_6
12


1/sqrt(494*56*293)
0.229487
0.24807
mixed_3x4
cyclo_3Xkrus_4
13
exp(2πi/53)
AK
1
 1/536
0.160022
0.23951
cyclic_13a
r=2, lambda=1
13
exp(2πi/169)
AK
1
1/1312
0.093701
0.23951
cyclic_13b
r=3
14
2cos(2π/29)
OK
2-θ
1/29(13/2) 
0.209425
0.23205
cyclo_14
 
15
2cos(2π/31)
OK
2-θ
1/317
0.201386
0.22548
cyclo_15
 
15
 
 
 
 1/(75*116)
0.200328
0.22548
mixed_3x5
cyclo_3Xcyclo_5
16
 
 
 
1/(54*177)
0.193613
0.21965
mixed_2x8
cyclo_2Xcyclo_8
16 2cos(2π/64)

1/2(79/2) 0.180648
0.21965 dast_16 [DAB02]
17
 exp(2πi/103)
AK
1
1/1038
0.112923
0.21444
cyclic_17a
r=5, lambda=26
17
 exp(2πi/289)
AK
1
1/1716
0.069491
0.21444
cyclic_17b
 r=3
18
2cos(2π/37)
OK
2-θ
 1/37(17/2)
0.181744
0.20973
cyclo_18
 
18
     
1/sqrt(59*1916)
0.180685
0.20973
mixed_3x6
 cyclo_3Xcyclo_6
19
 exp(2πi/191)
AK
1
1/1919 
0.083083
0.20547
cyclic_19a
r=19, lambda=138
20
2cos(2π/41)
OK
2-θ
1/41(19/2)
0.171367
0.20159
cyclo_20
 
21
2cos(2π/43)
OK
2-θ
1/4310
0.166785
0.19803
cyclo_21
 
21



1/sqrt(497*201343933)
0.157250
0.19803 mixed_3x7
krus_3Xkrus_7
22
 
 
 
1/sqrt(511*2320)
0.160801
0.19475
mixed_2x11
 cyclo_2Xcyclo_11
23
2cos(2π/47)
OK
2-θ
1/4711
0.158599
0.19173
cyclo_23
 
23
exp(2πi/47)
AK
1
1/4711
0.158599
0.19173
cyclic_23a
 as cyclo_23
r=5, lambda=12
24
 
 
 
1/sqrt(716*1721)
0.151348
0.18892
mixed_3x8
 cyclo_3Xcyclic_8
25
 
 
 
1/(520*1110)
0.105747
0.18631
mixed_5x5
 cyclo_5Xcyclic_5a
26
2cos(2π/53)
OK
2-θ
1/53(25/2)
0.148259
0.18388
cyclo_26
 
27
 
 
 
1/(79*1912)
0.141243
0.18160
 cyclo_3Xcyclo_9
28
 
 
 
1/(57*2913)
0.140051
0.17947
mixed_4x7
 krus_4Xkrus_7
29
2cos(2π/59)
OK
2-θ
1/5914
0.139670
0.17746
 cyclo_29
 
29
exp(2πi/59)
AK
1
1/5914
0.139670
0.17746
cyclic_29a
as cyclo_29
r=2, lambda=1
30
2cos(2π/61)
OK
2-θ
1/61(29/2)
0.137116
0.17556
cyclo_30
 
30
 
 
 
1/sqrt(1124*1325)
0.131613
0.17556 
mixed_5x6
 cyclo_5Xcyclo_6


 
 

Legend

n
 Dimension
θ
 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
α
 Twisting element in the ideal lattice construction
dp,min
 Minimum product distance
 dp,min1/n 
 Normalized minimum product distance
 (in red best known value, in boldface red optimal)
Bound
 Upper bound on dp,min1/n based on 
 Odlyzko's bound for the root discriminant
Matrix
 Generator matrix  in the row vector convention
Refs
 Bibliographic reference
OK
 Ring of integers of K
AK
 Ideal of OK such that AK2 is the codifferent of K

 
 
 

References

[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.

 
 
 
 
 
Last modified 16/3/2005 by Frederique Oggier and Emanuele Viterbo ©