November 2017, 11(4): 719-745. doi: 10.3934/amc.2017053

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

Received  April 2016 Revised  March 2017 Published  November 2017

Fund Project: X. Hou is supported by a Nanyang President Graduate Scholarship. The research of F. Oggier for this work is supported by Nanyang Technological University under Research Grant M58110049

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.

Citation: Xiaolu Hou, Frédérique Oggier. Modular lattices from a variation of construction a over number fields. Advances in Mathematics of Communications, 2017, 11 (4) : 719-745. doi: 10.3934/amc.2017053
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. BosmaJ. 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. ChapmanS. T. DoughertyP. 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. KositwattanarerkS. 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. LinF. 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. MallowsA. 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. OggierP. 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. BosmaJ. 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. ChapmanS. T. DoughertyP. 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. KositwattanarerkS. 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. LinF. 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. MallowsA. 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. OggierP. 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.

Table 1.  Weak Secrecy Gain-Dimension 8
No.Dim $d$ $\mu_{L}$ks $\chi_{L}^W$ $\Theta_{L}$
183281.2077108641201924245769201600
285281.0020108162496128208408480
38541201.29701000120024006000
4863161.1753100162448128144216400
587280.88381080246432128120192
6873161.104810016161680128224288
7811381.001510088824487288
8814280.530310802403282464
9814380.92161008083204880
10815380.88691008082406432
11815481.084010008160163264
12823380.6847100800240840
138235161.0396100001600160
14823581.1394100008082424
No.Dim $d$ $\mu_{L}$ks $\chi_{L}^W$ $\Theta_{L}$
183281.2077108641201924245769201600
285281.0020108162496128208408480
38541201.29701000120024006000
4863161.1753100162448128144216400
587280.88381080246432128120192
6873161.104810016161680128224288
7811381.001510088824487288
8814280.530310802403282464
9814380.92161008083204880
10815380.88691008082406432
11815481.084010008160163264
12823380.6847100800240840
138235161.0396100001600160
14823581.1394100008082424
Table 2.  Weak Secrecy Gain-Dimension 12
No.Dim $d$ $\mu_{L}$ks $\chi_{L}^W$ $\Theta_{L}$
151231120.469211260172396103225244704836417164
16123140.8342142810033298422365024977216516
17123140.9385141210042898420925024970816516
181232241.2012102464228960220051841052416192
191232121.3650101264300960209251841047616192
201233641.580610064372960198451841042816192
211252121.00301012246024040098421723440
221254601.604810006028852096019803680
231261120.182011260160252312556110417402796
24126160.384516205813223646093615642478
25126280.9797108203614426454412442016
261263161.358010016369625662413082112
271263121.3974100124010024466812842076
281263121.504410043613225666013081980
291271120.14521126016025231254497211641596
30127140.4645141232601684165808761684
31127140.5806144168415220858012681908
321272120.7584101216361441123848521056
33127280.879510816281121603847721152
34127341.402310043684643849721368
351211180.1765182436601803564246121204
361211140.217314164888152204144316772
3712113121.07261001201210872108436
381214180.133118243656148264320544912
391214140.153414164888152204144280628
4012143120.91341001200724872256
411215180.131318243232112292352328744
421215140.3899144012569680132388
431215120.46611201032304496128186
441215260.545510684424674136154
451215260.921710224242046100154
461215341.00311004818283664104
471215441.35731000410124872108
481215541.52651000041244108112
491223180.069818243656144228192316652
501223140.073514164888152204144280628
5112233120.569010012006000172
No.Dim $d$ $\mu_{L}$ks $\chi_{L}^W$ $\Theta_{L}$
151231120.469211260172396103225244704836417164
16123140.8342142810033298422365024977216516
17123140.9385141210042898420925024970816516
181232241.2012102464228960220051841052416192
191232121.3650101264300960209251841047616192
201233641.580610064372960198451841042816192
211252121.00301012246024040098421723440
221254601.604810006028852096019803680
231261120.182011260160252312556110417402796
24126160.384516205813223646093615642478
25126280.9797108203614426454412442016
261263161.358010016369625662413082112
271263121.3974100124010024466812842076
281263121.504410043613225666013081980
291271120.14521126016025231254497211641596
30127140.4645141232601684165808761684
31127140.5806144168415220858012681908
321272120.7584101216361441123848521056
33127280.879510816281121603847721152
34127341.402310043684643849721368
351211180.1765182436601803564246121204
361211140.217314164888152204144316772
3712113121.07261001201210872108436
381214180.133118243656148264320544912
391214140.153414164888152204144280628
4012143120.91341001200724872256
411215180.131318243232112292352328744
421215140.3899144012569680132388
431215120.46611201032304496128186
441215260.545510684424674136154
451215260.921710224242046100154
461215341.00311004818283664104
471215441.35731000410124872108
481215541.52651000041244108112
491223180.069818243656144228192316652
501223140.073514164888152204144280628
5112233120.569010012006000172
Table 3.  Weak Secrecy Gain-Dimension 16
No.Dim $d$ $\mu_{L}$ks $\chi_{L}^W$ $\Theta_{L}$
521632161.45851016128304140868641958447600112768
531632121.6669101248440180863321886447648113968
54163281.761210848416180864401886448016113968
55163241.830310464360172866761900848448113728
56165221.76711024722168842452643214520
5716542401.6822100024004800156000
5816541121.92131000112012482048587216384
591654641.98551000641928642432644814656
601654482.00791000482567362560664014080
611662160.8582101616112256560179229287616
621663181.56621001844122392105028967126
63166381.7693100832124376111230007156
64166381.8272100816120448112829927176
651673321.220610032323241676812163648
66167361.76041006127425256015363968
67167321.83811002168621249615564072
6816113161.098510016016176961921072
6916113161.1138100160121641002401092
7016143160.886410016001286496640
7116143160.8933100160012452100676
721615461.51871000610225478182
731615441.6192100044344074182
741615441.76601000401424134156
751615421.8018100024103884208
761615541.9146100004826100178
771615541.934410000443674170
781615521.8890100002164270160
7916233160.4715100160011200464
8016233160.4720100160011200460
No.Dim $d$ $\mu_{L}$ks $\chi_{L}^W$ $\Theta_{L}$
521632161.45851016128304140868641958447600112768
531632121.6669101248440180863321886447648113968
54163281.761210848416180864401886448016113968
55163241.830310464360172866761900848448113728
56165221.76711024722168842452643214520
5716542401.6822100024004800156000
5816541121.92131000112012482048587216384
591654641.98551000641928642432644814656
601654482.00791000482567362560664014080
611662160.8582101616112256560179229287616
621663181.56621001844122392105028967126
63166381.7693100832124376111230007156
64166381.8272100816120448112829927176
651673321.220610032323241676812163648
66167361.76041006127425256015363968
67167321.83811002168621249615564072
6816113161.098510016016176961921072
6916113161.1138100160121641002401092
7016143160.886410016001286496640
7116143160.8933100160012452100676
721615461.51871000610225478182
731615441.6192100044344074182
741615441.76601000401424134156
751615421.8018100024103884208
761615541.9146100004826100178
771615541.934410000443674170
781615521.8890100002164270160
7916233160.4715100160011200464
8016233160.4720100160011200460
Table 4.  Weak Secrecy Gain-Dimension 16 and minimum 3
No.Dim $d$ $\mu_{L}$ks $\chi_{L}^W$ $\Theta_{L}$
6916113161.1138100160121641002401092
6816113161.098510016016176961921072
7116143160.8933100160012452100676
7016143160.886410016001286496640
8016233160.4720100160011200460
7916233160.4715100160011200464
No.Dim $d$ $\mu_{L}$ks $\chi_{L}^W$ $\Theta_{L}$
6916113161.1138100160121641002401092
6816113161.098510016016176961921072
7116143160.8933100160012452100676
7016143160.886410016001286496640
8016233160.4720100160011200460
7916233160.4715100160011200464
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