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Modular lattices from a variation of construction a over number fields
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore |
We consider a variation of Construction A of lattices from linear codes based on two classes of number fields, totally real and CM Galois number fields. We propose a generic construction with explicit generator and Gram matrices, then focus on modular and unimodular lattices, obtained in the particular cases of totally real, respectively, imaginary, quadratic fields. Our motivation comes from coding theory, thus some relevant properties of modular lattices, such as minimal norm, theta series, kissing number and secrecy gain are analyzed. Interesting lattices are exhibited.
References:
[1] |
C. Bachoc,
Applications of coding theory to the construction of modular lattices, Journal of Combinatorial Theory, 78 (1997), 92-119.
doi: 10.1006/jcta.1996.2763. |
[2] |
E. Bayer-Fluckiger, Ideal lattices, A Panorama of Number Theory or The View from Baker's Garden, edited by Gisbert Wustholz Cambridge Univ. Press, Cambridge, (2002), 168-184. |
[3] |
E. Bayer-Fluckiger and I. Suarez,
Modular lattices over cyclotomic fields, Journal of Number Theory, 114 (2005), 394-411.
doi: 10.1016/j.jnt.2004.10.005. |
[4] |
V. Blomer,
Uniform bounds for Fourier coefficients of theta-series with arithmetic applications, Acta Arithmetica, 114 (2004), 1-21.
doi: 10.4064/aa114-1-1. |
[5] |
S. Böcherer and G. Nebe,
On theta series attached to maximal lattices and their adjoints, J.ramanujan Math.soc, 25 (2010), 265-284.
|
[6] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system. Ⅰ. The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[7] |
R. Chapman, S. T. Dougherty, P. Gaborit and P. Solé,
2-modular lattices from ternary codes, Journal De Théorie Des Nombres De Bordeaux, 14 (2002), 73-85.
doi: 10.5802/jtnb.347. |
[8] |
K. S. Chua and P. Solé,
Eisenstein lattices, Galois rings, and Theta Series, European Journal of Combinatorics, 25 (2004), 179-185.
doi: 10.1016/S0195-6698(03)00098-2. |
[9] |
J. H. Conway and N. J. A. Sloane,
Sphere Packings, Lattices and Groups, Springer, New York, 1988.
doi: 10. 1007/978-1-4757-6568-7. |
[10] |
J. H. Conway and N. J. A. Sloane,
A new upper bound for the minimum of an integral lattice of determinant one, Bull. Amer. Math. Soc., 23 (1990), 383-387.
doi: 10.1090/S0273-0979-1990-15940-3. |
[11] |
J. H. Conway and N. J. A. Sloane,
A note on optimal unimodular lattices, J. Number Theory, 72 (1998), 357-362.
doi: 10.1006/jnth.1998.2257. |
[12] |
W. Ebeling,
Lattices and Codes: A Course Partially Based on Lecturers by F. Hirzebruch Advanced Lectures in Mathematics, Springer, Germany, 2013.
doi: 10.1007/978-3-658-00360-9. |
[13] |
A.-M. Ernvall-Hytönen,
On a conjecture by Belfiore and Sol´e on some lattices, IEEE Trans. Inf. Theory, 58 (2012), 5950-5955.
doi: 10.1109/TIT.2012.2201915. |
[14] |
G. D. Forney,
Coset codes-part Ⅰ: Introduction and geometrical classification, IEEE Trans. Inform. Theory, 34 (1988), 1123-1151.
doi: 10.1109/18.21245. |
[15] |
X. Hou, F. Lin and F. Oggier, Construction and secrecy gain of a family of 5−modular lattices, in the proceedings of the IEEE Information Theory Workshop, (2014), 117-121.
doi: 10.1109/ITW.2014.6970804. |
[16] |
W. Kositwattanarerk, S. S. Ong and F. Oggier,
Construction a of lattices over number fields and block fading wiretap coding, IEEE Transactions on Information Theory, 61 (2015), 2273-2282.
doi: 10.1109/TIT.2015.2416340. |
[17] |
F. Lin and F. Oggier,
A classification of unimodular lattice wiretap codes in small dimensions, IEEE Trans. Inf. Theory, 59 (2013), 3295-3303.
doi: 10.1109/TIT.2013.2246814. |
[18] |
F. Lin, F. Oggier and P. Solé,
2-and 3-modular lattice wiretap codes in small dimensions, Applicable Algebra in Engineering, Communication and Computing, 26 (2015), 571-590.
doi: 10.1007/s00200-015-0267-2. |
[19] |
S. Ling and C. Xing,
Coding Theory: A First Course, Cambridge University Press, 2004.
doi: 10.1017/CBO9780511755279. |
[20] |
C. L. Mallows, A. M. Odlyzko and N. J. A. Sloane,
Upper bounds for modular forms, lattices and codes, Journal of Algebra, 36 (1975), 68-76.
doi: 10.1016/0021-8693(75)90155-6. |
[21] |
G. Nebe,
Finite subgroups of GL$_{24}(\mathbb{Q})$, Experimental Mathematics, 5 (1996), 163-195.
|
[22] |
G. Nebe,
Finite subgroups of GL$_n(\mathbb{Q})$ for $25≤n≤31$, Communications in Algebra, 24 (1996), 2341-2397.
|
[23] |
G. Nebe and K. Schindelar,
S-extremal strongly modular lattices, Journal de théorie des nombres de Bordeaux, 19 (2007), 683-701.
doi: 10.5802/jtnb.608. |
[24] |
G. Nebe,
Automorphisms of extremal unimodular lattices in dimension 72, Journal of Number Theory, 161 (2016), 362-383.
doi: 10.1016/j.jnt.2015.05.001. |
[25] |
J. Neukirch,
Algebraic Number Theory, Springer-Verlag, New York, 1999.
doi: 10.1007/978-3-662-03983-0. |
[26] |
F. Oggier and E. Viterbo,
Algebraic number theory and code design for Rayleigh fading channels, Foundations and Trends in Communications and Information Theory, 1 (2004), 333-415.
doi: 10.1561/0100000003. |
[27] |
F. Oggier, P. Solé and J.-C. Belfiore,
Lattice codes for the wiretap Gaussian channel: Construction and analysis, IEEE Transactions on Information Theory, 62 (2016), 5690-5708.
doi: 10.1109/TIT.2015.2494594. |
[28] |
F. Oggier and J. -C. Belfiore, Enabling multiplication in lattice codes via Construction A, in the proceedings of the IEEE Information Theory Workshop, 2013 (ITW), 9-13. |
[29] |
J. Pinchak and B. A. Sethuraman,
The Belfiore-Solé Conjecture and a certain technique for verifying it for a given lattice, Information Theory and Applications, (2014), 1-3.
doi: 10.1109/ITA.2014.6804279. |
[30] |
H.-G. Quebbemann,
Atkin-Lehner eigenforms and strongly modular lattices, L'Enseign. Math., 43 (1997), 55-65.
|
[31] |
H.-G. Quebbemann,
A shadow identity and an application to isoduality, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, 68 (1998), 339-345.
doi: 10.1007/BF02942571. |
[32] |
E. Rains and N. J. A. Sloane,
The shadow theory of modular and unimodular lattices, Journal of Number Theory, 73 (1999), 359-389.
doi: 10.1006/jnth.1998.2306. |
[33] |
SageMath, The Sage Mathematics Software System (Version 7. 1), The Sage Developers, 2016, http://www.sagemath.org. |
[34] |
N. J. A. Sloane,
Codes over GF(4) and complex lattices, Journal of Algebra, 52 (1978), 168-181.
doi: 10.1016/0021-8693(78)90266-1. |
[35] |
N. J. A. Sloane and G. Nebe, Catalogue of Lattices, published electronically at http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/. |
[36] |
Wolfram Research, Inc., Mathematica, Version 10. 4, Champaign, IL, 2016. |
show all references
References:
[1] |
C. Bachoc,
Applications of coding theory to the construction of modular lattices, Journal of Combinatorial Theory, 78 (1997), 92-119.
doi: 10.1006/jcta.1996.2763. |
[2] |
E. Bayer-Fluckiger, Ideal lattices, A Panorama of Number Theory or The View from Baker's Garden, edited by Gisbert Wustholz Cambridge Univ. Press, Cambridge, (2002), 168-184. |
[3] |
E. Bayer-Fluckiger and I. Suarez,
Modular lattices over cyclotomic fields, Journal of Number Theory, 114 (2005), 394-411.
doi: 10.1016/j.jnt.2004.10.005. |
[4] |
V. Blomer,
Uniform bounds for Fourier coefficients of theta-series with arithmetic applications, Acta Arithmetica, 114 (2004), 1-21.
doi: 10.4064/aa114-1-1. |
[5] |
S. Böcherer and G. Nebe,
On theta series attached to maximal lattices and their adjoints, J.ramanujan Math.soc, 25 (2010), 265-284.
|
[6] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system. Ⅰ. The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[7] |
R. Chapman, S. T. Dougherty, P. Gaborit and P. Solé,
2-modular lattices from ternary codes, Journal De Théorie Des Nombres De Bordeaux, 14 (2002), 73-85.
doi: 10.5802/jtnb.347. |
[8] |
K. S. Chua and P. Solé,
Eisenstein lattices, Galois rings, and Theta Series, European Journal of Combinatorics, 25 (2004), 179-185.
doi: 10.1016/S0195-6698(03)00098-2. |
[9] |
J. H. Conway and N. J. A. Sloane,
Sphere Packings, Lattices and Groups, Springer, New York, 1988.
doi: 10. 1007/978-1-4757-6568-7. |
[10] |
J. H. Conway and N. J. A. Sloane,
A new upper bound for the minimum of an integral lattice of determinant one, Bull. Amer. Math. Soc., 23 (1990), 383-387.
doi: 10.1090/S0273-0979-1990-15940-3. |
[11] |
J. H. Conway and N. J. A. Sloane,
A note on optimal unimodular lattices, J. Number Theory, 72 (1998), 357-362.
doi: 10.1006/jnth.1998.2257. |
[12] |
W. Ebeling,
Lattices and Codes: A Course Partially Based on Lecturers by F. Hirzebruch Advanced Lectures in Mathematics, Springer, Germany, 2013.
doi: 10.1007/978-3-658-00360-9. |
[13] |
A.-M. Ernvall-Hytönen,
On a conjecture by Belfiore and Sol´e on some lattices, IEEE Trans. Inf. Theory, 58 (2012), 5950-5955.
doi: 10.1109/TIT.2012.2201915. |
[14] |
G. D. Forney,
Coset codes-part Ⅰ: Introduction and geometrical classification, IEEE Trans. Inform. Theory, 34 (1988), 1123-1151.
doi: 10.1109/18.21245. |
[15] |
X. Hou, F. Lin and F. Oggier, Construction and secrecy gain of a family of 5−modular lattices, in the proceedings of the IEEE Information Theory Workshop, (2014), 117-121.
doi: 10.1109/ITW.2014.6970804. |
[16] |
W. Kositwattanarerk, S. S. Ong and F. Oggier,
Construction a of lattices over number fields and block fading wiretap coding, IEEE Transactions on Information Theory, 61 (2015), 2273-2282.
doi: 10.1109/TIT.2015.2416340. |
[17] |
F. Lin and F. Oggier,
A classification of unimodular lattice wiretap codes in small dimensions, IEEE Trans. Inf. Theory, 59 (2013), 3295-3303.
doi: 10.1109/TIT.2013.2246814. |
[18] |
F. Lin, F. Oggier and P. Solé,
2-and 3-modular lattice wiretap codes in small dimensions, Applicable Algebra in Engineering, Communication and Computing, 26 (2015), 571-590.
doi: 10.1007/s00200-015-0267-2. |
[19] |
S. Ling and C. Xing,
Coding Theory: A First Course, Cambridge University Press, 2004.
doi: 10.1017/CBO9780511755279. |
[20] |
C. L. Mallows, A. M. Odlyzko and N. J. A. Sloane,
Upper bounds for modular forms, lattices and codes, Journal of Algebra, 36 (1975), 68-76.
doi: 10.1016/0021-8693(75)90155-6. |
[21] |
G. Nebe,
Finite subgroups of GL$_{24}(\mathbb{Q})$, Experimental Mathematics, 5 (1996), 163-195.
|
[22] |
G. Nebe,
Finite subgroups of GL$_n(\mathbb{Q})$ for $25≤n≤31$, Communications in Algebra, 24 (1996), 2341-2397.
|
[23] |
G. Nebe and K. Schindelar,
S-extremal strongly modular lattices, Journal de théorie des nombres de Bordeaux, 19 (2007), 683-701.
doi: 10.5802/jtnb.608. |
[24] |
G. Nebe,
Automorphisms of extremal unimodular lattices in dimension 72, Journal of Number Theory, 161 (2016), 362-383.
doi: 10.1016/j.jnt.2015.05.001. |
[25] |
J. Neukirch,
Algebraic Number Theory, Springer-Verlag, New York, 1999.
doi: 10.1007/978-3-662-03983-0. |
[26] |
F. Oggier and E. Viterbo,
Algebraic number theory and code design for Rayleigh fading channels, Foundations and Trends in Communications and Information Theory, 1 (2004), 333-415.
doi: 10.1561/0100000003. |
[27] |
F. Oggier, P. Solé and J.-C. Belfiore,
Lattice codes for the wiretap Gaussian channel: Construction and analysis, IEEE Transactions on Information Theory, 62 (2016), 5690-5708.
doi: 10.1109/TIT.2015.2494594. |
[28] |
F. Oggier and J. -C. Belfiore, Enabling multiplication in lattice codes via Construction A, in the proceedings of the IEEE Information Theory Workshop, 2013 (ITW), 9-13. |
[29] |
J. Pinchak and B. A. Sethuraman,
The Belfiore-Solé Conjecture and a certain technique for verifying it for a given lattice, Information Theory and Applications, (2014), 1-3.
doi: 10.1109/ITA.2014.6804279. |
[30] |
H.-G. Quebbemann,
Atkin-Lehner eigenforms and strongly modular lattices, L'Enseign. Math., 43 (1997), 55-65.
|
[31] |
H.-G. Quebbemann,
A shadow identity and an application to isoduality, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, 68 (1998), 339-345.
doi: 10.1007/BF02942571. |
[32] |
E. Rains and N. J. A. Sloane,
The shadow theory of modular and unimodular lattices, Journal of Number Theory, 73 (1999), 359-389.
doi: 10.1006/jnth.1998.2306. |
[33] |
SageMath, The Sage Mathematics Software System (Version 7. 1), The Sage Developers, 2016, http://www.sagemath.org. |
[34] |
N. J. A. Sloane,
Codes over GF(4) and complex lattices, Journal of Algebra, 52 (1978), 168-181.
doi: 10.1016/0021-8693(78)90266-1. |
[35] |
N. J. A. Sloane and G. Nebe, Catalogue of Lattices, published electronically at http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/. |
[36] |
Wolfram Research, Inc., Mathematica, Version 10. 4, Champaign, IL, 2016. |
No. | Dim | | | ks | | | |||||||||
1 | 8 | 3 | 2 | 8 | 1.2077 | 1 | 0 | 8 | 64 | 120 | 192 | 424 | 576 | 920 | 1600 |
2 | 8 | 5 | 2 | 8 | 1.0020 | 1 | 0 | 8 | 16 | 24 | 96 | 128 | 208 | 408 | 480 |
3 | 8 | 5 | 4 | 120 | 1.2970 | 1 | 0 | 0 | 0 | 120 | 0 | 240 | 0 | 600 | 0 |
4 | 8 | 6 | 3 | 16 | 1.1753 | 1 | 0 | 0 | 16 | 24 | 48 | 128 | 144 | 216 | 400 |
5 | 8 | 7 | 2 | 8 | 0.8838 | 1 | 0 | 8 | 0 | 24 | 64 | 32 | 128 | 120 | 192 |
6 | 8 | 7 | 3 | 16 | 1.1048 | 1 | 0 | 0 | 16 | 16 | 16 | 80 | 128 | 224 | 288 |
7 | 8 | 11 | 3 | 8 | 1.0015 | 1 | 0 | 0 | 8 | 8 | 8 | 24 | 48 | 72 | 88 |
8 | 8 | 14 | 2 | 8 | 0.5303 | 1 | 0 | 8 | 0 | 24 | 0 | 32 | 8 | 24 | 64 |
9 | 8 | 14 | 3 | 8 | 0.9216 | 1 | 0 | 0 | 8 | 0 | 8 | 32 | 0 | 48 | 80 |
10 | 8 | 15 | 3 | 8 | 0.8869 | 1 | 0 | 0 | 8 | 0 | 8 | 24 | 0 | 64 | 32 |
11 | 8 | 15 | 4 | 8 | 1.0840 | 1 | 0 | 0 | 0 | 8 | 16 | 0 | 16 | 32 | 64 |
12 | 8 | 23 | 3 | 8 | 0.6847 | 1 | 0 | 0 | 8 | 0 | 0 | 24 | 0 | 8 | 40 |
13 | 8 | 23 | 5 | 16 | 1.0396 | 1 | 0 | 0 | 0 | 0 | 16 | 0 | 0 | 16 | 0 |
14 | 8 | 23 | 5 | 8 | 1.1394 | 1 | 0 | 0 | 0 | 0 | 8 | 0 | 8 | 24 | 24 |
No. | Dim | | | ks | | | |||||||||
1 | 8 | 3 | 2 | 8 | 1.2077 | 1 | 0 | 8 | 64 | 120 | 192 | 424 | 576 | 920 | 1600 |
2 | 8 | 5 | 2 | 8 | 1.0020 | 1 | 0 | 8 | 16 | 24 | 96 | 128 | 208 | 408 | 480 |
3 | 8 | 5 | 4 | 120 | 1.2970 | 1 | 0 | 0 | 0 | 120 | 0 | 240 | 0 | 600 | 0 |
4 | 8 | 6 | 3 | 16 | 1.1753 | 1 | 0 | 0 | 16 | 24 | 48 | 128 | 144 | 216 | 400 |
5 | 8 | 7 | 2 | 8 | 0.8838 | 1 | 0 | 8 | 0 | 24 | 64 | 32 | 128 | 120 | 192 |
6 | 8 | 7 | 3 | 16 | 1.1048 | 1 | 0 | 0 | 16 | 16 | 16 | 80 | 128 | 224 | 288 |
7 | 8 | 11 | 3 | 8 | 1.0015 | 1 | 0 | 0 | 8 | 8 | 8 | 24 | 48 | 72 | 88 |
8 | 8 | 14 | 2 | 8 | 0.5303 | 1 | 0 | 8 | 0 | 24 | 0 | 32 | 8 | 24 | 64 |
9 | 8 | 14 | 3 | 8 | 0.9216 | 1 | 0 | 0 | 8 | 0 | 8 | 32 | 0 | 48 | 80 |
10 | 8 | 15 | 3 | 8 | 0.8869 | 1 | 0 | 0 | 8 | 0 | 8 | 24 | 0 | 64 | 32 |
11 | 8 | 15 | 4 | 8 | 1.0840 | 1 | 0 | 0 | 0 | 8 | 16 | 0 | 16 | 32 | 64 |
12 | 8 | 23 | 3 | 8 | 0.6847 | 1 | 0 | 0 | 8 | 0 | 0 | 24 | 0 | 8 | 40 |
13 | 8 | 23 | 5 | 16 | 1.0396 | 1 | 0 | 0 | 0 | 0 | 16 | 0 | 0 | 16 | 0 |
14 | 8 | 23 | 5 | 8 | 1.1394 | 1 | 0 | 0 | 0 | 0 | 8 | 0 | 8 | 24 | 24 |
No. | Dim | | | ks | | | |||||||||
15 | 12 | 3 | 1 | 12 | 0.4692 | 1 | 12 | 60 | 172 | 396 | 1032 | 2524 | 4704 | 8364 | 17164 |
16 | 12 | 3 | 1 | 4 | 0.8342 | 1 | 4 | 28 | 100 | 332 | 984 | 2236 | 5024 | 9772 | 16516 |
17 | 12 | 3 | 1 | 4 | 0.9385 | 1 | 4 | 12 | 100 | 428 | 984 | 2092 | 5024 | 9708 | 16516 |
18 | 12 | 3 | 2 | 24 | 1.2012 | 1 | 0 | 24 | 64 | 228 | 960 | 2200 | 5184 | 10524 | 16192 |
19 | 12 | 3 | 2 | 12 | 1.3650 | 1 | 0 | 12 | 64 | 300 | 960 | 2092 | 5184 | 10476 | 16192 |
20 | 12 | 3 | 3 | 64 | 1.5806 | 1 | 0 | 0 | 64 | 372 | 960 | 1984 | 5184 | 10428 | 16192 |
21 | 12 | 5 | 2 | 12 | 1.0030 | 1 | 0 | 12 | 24 | 60 | 240 | 400 | 984 | 2172 | 3440 |
22 | 12 | 5 | 4 | 60 | 1.6048 | 1 | 0 | 0 | 0 | 60 | 288 | 520 | 960 | 1980 | 3680 |
23 | 12 | 6 | 1 | 12 | 0.1820 | 1 | 12 | 60 | 160 | 252 | 312 | 556 | 1104 | 1740 | 2796 |
24 | 12 | 6 | 1 | 6 | 0.3845 | 1 | 6 | 20 | 58 | 132 | 236 | 460 | 936 | 1564 | 2478 |
25 | 12 | 6 | 2 | 8 | 0.9797 | 1 | 0 | 8 | 20 | 36 | 144 | 264 | 544 | 1244 | 2016 |
26 | 12 | 6 | 3 | 16 | 1.3580 | 1 | 0 | 0 | 16 | 36 | 96 | 256 | 624 | 1308 | 2112 |
27 | 12 | 6 | 3 | 12 | 1.3974 | 1 | 0 | 0 | 12 | 40 | 100 | 244 | 668 | 1284 | 2076 |
28 | 12 | 6 | 3 | 12 | 1.5044 | 1 | 0 | 0 | 4 | 36 | 132 | 256 | 660 | 1308 | 1980 |
29 | 12 | 7 | 1 | 12 | 0.1452 | 1 | 12 | 60 | 160 | 252 | 312 | 544 | 972 | 1164 | 1596 |
30 | 12 | 7 | 1 | 4 | 0.4645 | 1 | 4 | 12 | 32 | 60 | 168 | 416 | 580 | 876 | 1684 |
31 | 12 | 7 | 1 | 4 | 0.5806 | 1 | 4 | 4 | 16 | 84 | 152 | 208 | 580 | 1268 | 1908 |
32 | 12 | 7 | 2 | 12 | 0.7584 | 1 | 0 | 12 | 16 | 36 | 144 | 112 | 384 | 852 | 1056 |
33 | 12 | 7 | 2 | 8 | 0.8795 | 1 | 0 | 8 | 16 | 28 | 112 | 160 | 384 | 772 | 1152 |
34 | 12 | 7 | 3 | 4 | 1.4023 | 1 | 0 | 0 | 4 | 36 | 84 | 64 | 384 | 972 | 1368 |
35 | 12 | 11 | 1 | 8 | 0.1765 | 1 | 8 | 24 | 36 | 60 | 180 | 356 | 424 | 612 | 1204 |
36 | 12 | 11 | 1 | 4 | 0.2173 | 1 | 4 | 16 | 48 | 88 | 152 | 204 | 144 | 316 | 772 |
37 | 12 | 11 | 3 | 12 | 1.0726 | 1 | 0 | 0 | 12 | 0 | 12 | 108 | 72 | 108 | 436 |
38 | 12 | 14 | 1 | 8 | 0.1331 | 1 | 8 | 24 | 36 | 56 | 148 | 264 | 320 | 544 | 912 |
39 | 12 | 14 | 1 | 4 | 0.1534 | 1 | 4 | 16 | 48 | 88 | 152 | 204 | 144 | 280 | 628 |
40 | 12 | 14 | 3 | 12 | 0.9134 | 1 | 0 | 0 | 12 | 0 | 0 | 72 | 48 | 72 | 256 |
41 | 12 | 15 | 1 | 8 | 0.1313 | 1 | 8 | 24 | 32 | 32 | 112 | 292 | 352 | 328 | 744 |
42 | 12 | 15 | 1 | 4 | 0.3899 | 1 | 4 | 4 | 0 | 12 | 56 | 96 | 80 | 132 | 388 |
43 | 12 | 15 | 1 | 2 | 0.4661 | 1 | 2 | 0 | 10 | 32 | 30 | 44 | 96 | 128 | 186 |
44 | 12 | 15 | 2 | 6 | 0.5455 | 1 | 0 | 6 | 8 | 4 | 42 | 46 | 74 | 136 | 154 |
45 | 12 | 15 | 2 | 6 | 0.9217 | 1 | 0 | 2 | 2 | 4 | 24 | 20 | 46 | 100 | 154 |
46 | 12 | 15 | 3 | 4 | 1.0031 | 1 | 0 | 0 | 4 | 8 | 18 | 28 | 36 | 64 | 104 |
47 | 12 | 15 | 4 | 4 | 1.3573 | 1 | 0 | 0 | 0 | 4 | 10 | 12 | 48 | 72 | 108 |
48 | 12 | 15 | 5 | 4 | 1.5265 | 1 | 0 | 0 | 0 | 0 | 4 | 12 | 44 | 108 | 112 |
49 | 12 | 23 | 1 | 8 | 0.0698 | 1 | 8 | 24 | 36 | 56 | 144 | 228 | 192 | 316 | 652 |
50 | 12 | 23 | 1 | 4 | 0.0735 | 1 | 4 | 16 | 48 | 88 | 152 | 204 | 144 | 280 | 628 |
51 | 12 | 23 | 3 | 12 | 0.5690 | 1 | 0 | 0 | 12 | 0 | 0 | 60 | 0 | 0 | 172 |
No. | Dim | | | ks | | | |||||||||
15 | 12 | 3 | 1 | 12 | 0.4692 | 1 | 12 | 60 | 172 | 396 | 1032 | 2524 | 4704 | 8364 | 17164 |
16 | 12 | 3 | 1 | 4 | 0.8342 | 1 | 4 | 28 | 100 | 332 | 984 | 2236 | 5024 | 9772 | 16516 |
17 | 12 | 3 | 1 | 4 | 0.9385 | 1 | 4 | 12 | 100 | 428 | 984 | 2092 | 5024 | 9708 | 16516 |
18 | 12 | 3 | 2 | 24 | 1.2012 | 1 | 0 | 24 | 64 | 228 | 960 | 2200 | 5184 | 10524 | 16192 |
19 | 12 | 3 | 2 | 12 | 1.3650 | 1 | 0 | 12 | 64 | 300 | 960 | 2092 | 5184 | 10476 | 16192 |
20 | 12 | 3 | 3 | 64 | 1.5806 | 1 | 0 | 0 | 64 | 372 | 960 | 1984 | 5184 | 10428 | 16192 |
21 | 12 | 5 | 2 | 12 | 1.0030 | 1 | 0 | 12 | 24 | 60 | 240 | 400 | 984 | 2172 | 3440 |
22 | 12 | 5 | 4 | 60 | 1.6048 | 1 | 0 | 0 | 0 | 60 | 288 | 520 | 960 | 1980 | 3680 |
23 | 12 | 6 | 1 | 12 | 0.1820 | 1 | 12 | 60 | 160 | 252 | 312 | 556 | 1104 | 1740 | 2796 |
24 | 12 | 6 | 1 | 6 | 0.3845 | 1 | 6 | 20 | 58 | 132 | 236 | 460 | 936 | 1564 | 2478 |
25 | 12 | 6 | 2 | 8 | 0.9797 | 1 | 0 | 8 | 20 | 36 | 144 | 264 | 544 | 1244 | 2016 |
26 | 12 | 6 | 3 | 16 | 1.3580 | 1 | 0 | 0 | 16 | 36 | 96 | 256 | 624 | 1308 | 2112 |
27 | 12 | 6 | 3 | 12 | 1.3974 | 1 | 0 | 0 | 12 | 40 | 100 | 244 | 668 | 1284 | 2076 |
28 | 12 | 6 | 3 | 12 | 1.5044 | 1 | 0 | 0 | 4 | 36 | 132 | 256 | 660 | 1308 | 1980 |
29 | 12 | 7 | 1 | 12 | 0.1452 | 1 | 12 | 60 | 160 | 252 | 312 | 544 | 972 | 1164 | 1596 |
30 | 12 | 7 | 1 | 4 | 0.4645 | 1 | 4 | 12 | 32 | 60 | 168 | 416 | 580 | 876 | 1684 |
31 | 12 | 7 | 1 | 4 | 0.5806 | 1 | 4 | 4 | 16 | 84 | 152 | 208 | 580 | 1268 | 1908 |
32 | 12 | 7 | 2 | 12 | 0.7584 | 1 | 0 | 12 | 16 | 36 | 144 | 112 | 384 | 852 | 1056 |
33 | 12 | 7 | 2 | 8 | 0.8795 | 1 | 0 | 8 | 16 | 28 | 112 | 160 | 384 | 772 | 1152 |
34 | 12 | 7 | 3 | 4 | 1.4023 | 1 | 0 | 0 | 4 | 36 | 84 | 64 | 384 | 972 | 1368 |
35 | 12 | 11 | 1 | 8 | 0.1765 | 1 | 8 | 24 | 36 | 60 | 180 | 356 | 424 | 612 | 1204 |
36 | 12 | 11 | 1 | 4 | 0.2173 | 1 | 4 | 16 | 48 | 88 | 152 | 204 | 144 | 316 | 772 |
37 | 12 | 11 | 3 | 12 | 1.0726 | 1 | 0 | 0 | 12 | 0 | 12 | 108 | 72 | 108 | 436 |
38 | 12 | 14 | 1 | 8 | 0.1331 | 1 | 8 | 24 | 36 | 56 | 148 | 264 | 320 | 544 | 912 |
39 | 12 | 14 | 1 | 4 | 0.1534 | 1 | 4 | 16 | 48 | 88 | 152 | 204 | 144 | 280 | 628 |
40 | 12 | 14 | 3 | 12 | 0.9134 | 1 | 0 | 0 | 12 | 0 | 0 | 72 | 48 | 72 | 256 |
41 | 12 | 15 | 1 | 8 | 0.1313 | 1 | 8 | 24 | 32 | 32 | 112 | 292 | 352 | 328 | 744 |
42 | 12 | 15 | 1 | 4 | 0.3899 | 1 | 4 | 4 | 0 | 12 | 56 | 96 | 80 | 132 | 388 |
43 | 12 | 15 | 1 | 2 | 0.4661 | 1 | 2 | 0 | 10 | 32 | 30 | 44 | 96 | 128 | 186 |
44 | 12 | 15 | 2 | 6 | 0.5455 | 1 | 0 | 6 | 8 | 4 | 42 | 46 | 74 | 136 | 154 |
45 | 12 | 15 | 2 | 6 | 0.9217 | 1 | 0 | 2 | 2 | 4 | 24 | 20 | 46 | 100 | 154 |
46 | 12 | 15 | 3 | 4 | 1.0031 | 1 | 0 | 0 | 4 | 8 | 18 | 28 | 36 | 64 | 104 |
47 | 12 | 15 | 4 | 4 | 1.3573 | 1 | 0 | 0 | 0 | 4 | 10 | 12 | 48 | 72 | 108 |
48 | 12 | 15 | 5 | 4 | 1.5265 | 1 | 0 | 0 | 0 | 0 | 4 | 12 | 44 | 108 | 112 |
49 | 12 | 23 | 1 | 8 | 0.0698 | 1 | 8 | 24 | 36 | 56 | 144 | 228 | 192 | 316 | 652 |
50 | 12 | 23 | 1 | 4 | 0.0735 | 1 | 4 | 16 | 48 | 88 | 152 | 204 | 144 | 280 | 628 |
51 | 12 | 23 | 3 | 12 | 0.5690 | 1 | 0 | 0 | 12 | 0 | 0 | 60 | 0 | 0 | 172 |
No. | Dim | | | ks | | | |||||||||
52 | 16 | 3 | 2 | 16 | 1.4585 | 1 | 0 | 16 | 128 | 304 | 1408 | 6864 | 19584 | 47600 | 112768 |
53 | 16 | 3 | 2 | 12 | 1.6669 | 1 | 0 | 12 | 48 | 440 | 1808 | 6332 | 18864 | 47648 | 113968 |
54 | 16 | 3 | 2 | 8 | 1.7612 | 1 | 0 | 8 | 48 | 416 | 1808 | 6440 | 18864 | 48016 | 113968 |
55 | 16 | 3 | 2 | 4 | 1.8303 | 1 | 0 | 4 | 64 | 360 | 1728 | 6676 | 19008 | 48448 | 113728 |
56 | 16 | 5 | 2 | 2 | 1.7671 | 1 | 0 | 2 | 4 | 72 | 216 | 884 | 2452 | 6432 | 14520 |
57 | 16 | 5 | 4 | 240 | 1.6822 | 1 | 0 | 0 | 0 | 240 | 0 | 480 | 0 | 15600 | 0 |
58 | 16 | 5 | 4 | 112 | 1.9213 | 1 | 0 | 0 | 0 | 112 | 0 | 1248 | 2048 | 5872 | 16384 |
59 | 16 | 5 | 4 | 64 | 1.9855 | 1 | 0 | 0 | 0 | 64 | 192 | 864 | 2432 | 6448 | 14656 |
60 | 16 | 5 | 4 | 48 | 2.0079 | 1 | 0 | 0 | 0 | 48 | 256 | 736 | 2560 | 6640 | 14080 |
61 | 16 | 6 | 2 | 16 | 0.8582 | 1 | 0 | 16 | 16 | 112 | 256 | 560 | 1792 | 2928 | 7616 |
62 | 16 | 6 | 3 | 18 | 1.5662 | 1 | 0 | 0 | 18 | 44 | 122 | 392 | 1050 | 2896 | 7126 |
63 | 16 | 6 | 3 | 8 | 1.7693 | 1 | 0 | 0 | 8 | 32 | 124 | 376 | 1112 | 3000 | 7156 |
64 | 16 | 6 | 3 | 8 | 1.8272 | 1 | 0 | 0 | 8 | 16 | 120 | 448 | 1128 | 2992 | 7176 |
65 | 16 | 7 | 3 | 32 | 1.2206 | 1 | 0 | 0 | 32 | 32 | 32 | 416 | 768 | 1216 | 3648 |
66 | 16 | 7 | 3 | 6 | 1.7604 | 1 | 0 | 0 | 6 | 12 | 74 | 252 | 560 | 1536 | 3968 |
67 | 16 | 7 | 3 | 2 | 1.8381 | 1 | 0 | 0 | 2 | 16 | 86 | 212 | 496 | 1556 | 4072 |
68 | 16 | 11 | 3 | 16 | 1.0985 | 1 | 0 | 0 | 16 | 0 | 16 | 176 | 96 | 192 | 1072 |
69 | 16 | 11 | 3 | 16 | 1.1138 | 1 | 0 | 0 | 16 | 0 | 12 | 164 | 100 | 240 | 1092 |
70 | 16 | 14 | 3 | 16 | 0.8864 | 1 | 0 | 0 | 16 | 0 | 0 | 128 | 64 | 96 | 640 |
71 | 16 | 14 | 3 | 16 | 0.8933 | 1 | 0 | 0 | 16 | 0 | 0 | 124 | 52 | 100 | 676 |
72 | 16 | 15 | 4 | 6 | 1.5187 | 1 | 0 | 0 | 0 | 6 | 10 | 22 | 54 | 78 | 182 |
73 | 16 | 15 | 4 | 4 | 1.6192 | 1 | 0 | 0 | 0 | 4 | 4 | 34 | 40 | 74 | 182 |
74 | 16 | 15 | 4 | 4 | 1.7660 | 1 | 0 | 0 | 0 | 4 | 0 | 14 | 24 | 134 | 156 |
75 | 16 | 15 | 4 | 2 | 1.8018 | 1 | 0 | 0 | 0 | 2 | 4 | 10 | 38 | 84 | 208 |
76 | 16 | 15 | 5 | 4 | 1.9146 | 1 | 0 | 0 | 0 | 0 | 4 | 8 | 26 | 100 | 178 |
77 | 16 | 15 | 5 | 4 | 1.9344 | 1 | 0 | 0 | 0 | 0 | 4 | 4 | 36 | 74 | 170 |
78 | 16 | 15 | 5 | 2 | 1.8890 | 1 | 0 | 0 | 0 | 0 | 2 | 16 | 42 | 70 | 160 |
79 | 16 | 23 | 3 | 16 | 0.4715 | 1 | 0 | 0 | 16 | 0 | 0 | 112 | 0 | 0 | 464 |
80 | 16 | 23 | 3 | 16 | 0.4720 | 1 | 0 | 0 | 16 | 0 | 0 | 112 | 0 | 0 | 460 |
No. | Dim | | | ks | | | |||||||||
52 | 16 | 3 | 2 | 16 | 1.4585 | 1 | 0 | 16 | 128 | 304 | 1408 | 6864 | 19584 | 47600 | 112768 |
53 | 16 | 3 | 2 | 12 | 1.6669 | 1 | 0 | 12 | 48 | 440 | 1808 | 6332 | 18864 | 47648 | 113968 |
54 | 16 | 3 | 2 | 8 | 1.7612 | 1 | 0 | 8 | 48 | 416 | 1808 | 6440 | 18864 | 48016 | 113968 |
55 | 16 | 3 | 2 | 4 | 1.8303 | 1 | 0 | 4 | 64 | 360 | 1728 | 6676 | 19008 | 48448 | 113728 |
56 | 16 | 5 | 2 | 2 | 1.7671 | 1 | 0 | 2 | 4 | 72 | 216 | 884 | 2452 | 6432 | 14520 |
57 | 16 | 5 | 4 | 240 | 1.6822 | 1 | 0 | 0 | 0 | 240 | 0 | 480 | 0 | 15600 | 0 |
58 | 16 | 5 | 4 | 112 | 1.9213 | 1 | 0 | 0 | 0 | 112 | 0 | 1248 | 2048 | 5872 | 16384 |
59 | 16 | 5 | 4 | 64 | 1.9855 | 1 | 0 | 0 | 0 | 64 | 192 | 864 | 2432 | 6448 | 14656 |
60 | 16 | 5 | 4 | 48 | 2.0079 | 1 | 0 | 0 | 0 | 48 | 256 | 736 | 2560 | 6640 | 14080 |
61 | 16 | 6 | 2 | 16 | 0.8582 | 1 | 0 | 16 | 16 | 112 | 256 | 560 | 1792 | 2928 | 7616 |
62 | 16 | 6 | 3 | 18 | 1.5662 | 1 | 0 | 0 | 18 | 44 | 122 | 392 | 1050 | 2896 | 7126 |
63 | 16 | 6 | 3 | 8 | 1.7693 | 1 | 0 | 0 | 8 | 32 | 124 | 376 | 1112 | 3000 | 7156 |
64 | 16 | 6 | 3 | 8 | 1.8272 | 1 | 0 | 0 | 8 | 16 | 120 | 448 | 1128 | 2992 | 7176 |
65 | 16 | 7 | 3 | 32 | 1.2206 | 1 | 0 | 0 | 32 | 32 | 32 | 416 | 768 | 1216 | 3648 |
66 | 16 | 7 | 3 | 6 | 1.7604 | 1 | 0 | 0 | 6 | 12 | 74 | 252 | 560 | 1536 | 3968 |
67 | 16 | 7 | 3 | 2 | 1.8381 | 1 | 0 | 0 | 2 | 16 | 86 | 212 | 496 | 1556 | 4072 |
68 | 16 | 11 | 3 | 16 | 1.0985 | 1 | 0 | 0 | 16 | 0 | 16 | 176 | 96 | 192 | 1072 |
69 | 16 | 11 | 3 | 16 | 1.1138 | 1 | 0 | 0 | 16 | 0 | 12 | 164 | 100 | 240 | 1092 |
70 | 16 | 14 | 3 | 16 | 0.8864 | 1 | 0 | 0 | 16 | 0 | 0 | 128 | 64 | 96 | 640 |
71 | 16 | 14 | 3 | 16 | 0.8933 | 1 | 0 | 0 | 16 | 0 | 0 | 124 | 52 | 100 | 676 |
72 | 16 | 15 | 4 | 6 | 1.5187 | 1 | 0 | 0 | 0 | 6 | 10 | 22 | 54 | 78 | 182 |
73 | 16 | 15 | 4 | 4 | 1.6192 | 1 | 0 | 0 | 0 | 4 | 4 | 34 | 40 | 74 | 182 |
74 | 16 | 15 | 4 | 4 | 1.7660 | 1 | 0 | 0 | 0 | 4 | 0 | 14 | 24 | 134 | 156 |
75 | 16 | 15 | 4 | 2 | 1.8018 | 1 | 0 | 0 | 0 | 2 | 4 | 10 | 38 | 84 | 208 |
76 | 16 | 15 | 5 | 4 | 1.9146 | 1 | 0 | 0 | 0 | 0 | 4 | 8 | 26 | 100 | 178 |
77 | 16 | 15 | 5 | 4 | 1.9344 | 1 | 0 | 0 | 0 | 0 | 4 | 4 | 36 | 74 | 170 |
78 | 16 | 15 | 5 | 2 | 1.8890 | 1 | 0 | 0 | 0 | 0 | 2 | 16 | 42 | 70 | 160 |
79 | 16 | 23 | 3 | 16 | 0.4715 | 1 | 0 | 0 | 16 | 0 | 0 | 112 | 0 | 0 | 464 |
80 | 16 | 23 | 3 | 16 | 0.4720 | 1 | 0 | 0 | 16 | 0 | 0 | 112 | 0 | 0 | 460 |
No. | Dim | | | ks | | | |||||||||
69 | 16 | 11 | 3 | 16 | 1.1138 | 1 | 0 | 0 | 16 | 0 | 12 | 164 | 100 | 240 | 1092 |
68 | 16 | 11 | 3 | 16 | 1.0985 | 1 | 0 | 0 | 16 | 0 | 16 | 176 | 96 | 192 | 1072 |
71 | 16 | 14 | 3 | 16 | 0.8933 | 1 | 0 | 0 | 16 | 0 | 0 | 124 | 52 | 100 | 676 |
70 | 16 | 14 | 3 | 16 | 0.8864 | 1 | 0 | 0 | 16 | 0 | 0 | 128 | 64 | 96 | 640 |
80 | 16 | 23 | 3 | 16 | 0.4720 | 1 | 0 | 0 | 16 | 0 | 0 | 112 | 0 | 0 | 460 |
79 | 16 | 23 | 3 | 16 | 0.4715 | 1 | 0 | 0 | 16 | 0 | 0 | 112 | 0 | 0 | 464 |
No. | Dim | | | ks | | | |||||||||
69 | 16 | 11 | 3 | 16 | 1.1138 | 1 | 0 | 0 | 16 | 0 | 12 | 164 | 100 | 240 | 1092 |
68 | 16 | 11 | 3 | 16 | 1.0985 | 1 | 0 | 0 | 16 | 0 | 16 | 176 | 96 | 192 | 1072 |
71 | 16 | 14 | 3 | 16 | 0.8933 | 1 | 0 | 0 | 16 | 0 | 0 | 124 | 52 | 100 | 676 |
70 | 16 | 14 | 3 | 16 | 0.8864 | 1 | 0 | 0 | 16 | 0 | 0 | 128 | 64 | 96 | 640 |
80 | 16 | 23 | 3 | 16 | 0.4720 | 1 | 0 | 0 | 16 | 0 | 0 | 112 | 0 | 0 | 460 |
79 | 16 | 23 | 3 | 16 | 0.4715 | 1 | 0 | 0 | 16 | 0 | 0 | 112 | 0 | 0 | 464 |
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