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doi: 10.3934/amc.2020118

Rotated $ A_n $-lattice codes of full diversity

Agnaldo José Ferrari , and  Tatiana Miguel Rodrigues de Souza

School of Sciences, Department of Mathematics, São Paulo State University - UNESP, Bauru, SP 17033-360, BR

* Corresponding author

Received  August 2020 Revised  September 2020 Published  November 2020

Fund Project: This work was supported by FAPESP 2013/25977-7 and CNPq 429346/2018-2

Some important properties of lattices are packing density and full diversity, which may be good for signal transmission over both Gaussian and Rayleigh fading channel, respectively. The algebraic lattices are constructed through twisted homomorphism of some modules in the ring of integers of a number field $ \mathbb{K} $. In this paper, we present a construction of some families of rotated $ A_n- $lattices, for $ n = 2^{r-2}-1 $, $ r \geq 4 $, via totally real subfield of cyclotomic fields. Furthermore, closed-form expressions for the minimum product distance of those lattices are obtained through algebraic properties.

Keywords: Algebraic number field, algebraic lattice, packing density, twisted homomorphism.
Mathematics Subject Classification: Primary: 52C07; Secondary: 11H31, 11H71.
Citation: Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $ A_n $-lattice codes of full diversity. Advances in Mathematics of Communications, doi: 10.3934/amc.2020118
References:
[1]

E. Bayer-Fluckiger, Ideal lattices, in A Panorama of Number Theory or the View from Baker's Garden, Cambridge Univ. Press, Cambridge, 2002,165-184. doi: 10.1017/CBO9780511542961.012.  Google Scholar

[2]

E. Bayer-Fluckiger, Lattices and number fields, in Algebraic Geometry: Hirzebruch 70, Contemp. Math., 241, Amer. Math. Soc., Providence, RI, 1999, 69–84. doi: 10.1090/conm/241/03628.  Google Scholar

[3]

E. Bayer-Fluckiger, Upper bounds for Euclidean minima of algebraic number fields, J. Number Theory, 121 (2006), 305-323.  doi: 10.1016/j.jnt.2006.03.002.  Google Scholar

[4]

E. Bayer-Fluckiger and G. Nebe, On the Euclidian minimum of some real number fields, J. Théor. Nombres Bordeaux, 17 (2005), 437–454. doi: 10.5802/jtnb.500.  Google Scholar

[5]

E. Bayer-Fluckiger, F. Oggier and E. Viterbo, New algebraic constructions of rotated $\mathbb{Z}^n$-lattice constellations for the Rayleigh fading channel, IEEE Trans. Inform. Theory, 50 (2004), 702–714. doi: 10.1109/TIT.2004.825045.  Google Scholar

[6]

E. Bayer-Fluckiger and I. Suarez, Ideal lattices over totally real number fields and Euclidean minima, Arch. Math. (Basel), 86 (2006), 217–225. doi: 10.1007/s00013-005-1469-9.  Google Scholar

[7]

K. Boullé and J. C. Belfiore, Modulation scheme design for the Rayleigh fading channel, Proc. Conf. Information Science and System, (1992), 288–293. Google Scholar

[8]

J. Boutros, E. Viterbo, C. Rastello and J.-C. Belfiore, Good lattice constellations for both Rayleigh fading and Gaussian channels, IEEE Trans. Inform. Theory, 42 (1996), 502–518. doi: 10.1109/18.485720.  Google Scholar

[9]

H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, Ann. of Math. (2), 170 (2009), 1003–1050. doi: 10.4007/annals.2009.170.1003.  Google Scholar

[10]

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Fundamental Principles of Mathematical Sciences, 290, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4757-2249-9.  Google Scholar

[11]

J. H. Conway and N. J. A. Sloane, The optimal isodual lattice quantizer in three dimensions, Adv. Math. Commun., 1 (2007), 257–260. doi: 10.3934/amc.2007.1.257.  Google Scholar

[12]

P. Elia, B. A. Sethuraman and P. V. Kumar, Perfect space-time codes for any number of antennas, IEEE Trans. Inform. Theory, 53 (2007), 3853–3868. doi: 10.1109/TIT.2007.907502.  Google Scholar

[13]

X. Hou and F. Oggier, Modular lattices from a variation of Construction A over number fields, Adv. Math. Commun., 11 (2017), 719–745. doi: 10.3934/amc.2017053.  Google Scholar

[14]

G. C. Jorge, A. A. de Andrade, S. I. R. Costa and J. E. Strapasson, Algebraic constructions of densest lattices, J. Algebra, 429 (2015), 218–235. doi: 10.1016/j.jalgebra.2014.12.044.  Google Scholar

[15]

G. C. Jorge and S. I. R. Costa, On rotated $D_n$-lattices constructed via totally real number fields, Arch. Math. (Basel), 100 (2013), 323–332. doi: 10.1007/s00013-013-0501-8.  Google Scholar

[16]

G. C. Jorge, A. J. Ferrari and S. I. R. Costa, Rotated $D_n$-lattices, J. Number Theory, 132 (2012), 2397–2406. doi: 10.1016/j.jnt.2012.05.002.  Google Scholar

[17]

D. Micciancio and S. Goldwasser, Complexity of Lattice Problems. A Cryptographic Perspective, The Kluwer International Series in Engineering and Computer Science, 671, Kluwer Academic Publishers, Boston, MA, 2002. doi: 10.1007/978-1-4615-0897-7.  Google Scholar

[18]

F. Oggier, Algebraic Methods for Channel Coding, Ph.D Thesis, École Polytechnique Fédérale in Lausanne, Lausanne, 2005. Google Scholar

[19]

F. Oggier and E. Bayer-Fluckiger, Best rotated cubic lattice constellations for the Rayleigh fading channel, Proceedings of IEEE International Symposium on Information Theory, Yokohama, Japan, 2003. Google Scholar

[20]

P. Samuel, Algebraic Theory of Numbers, Houghton Mifflin Co., Boston, MA, 1970,109pp.  Google Scholar

[21]

L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, 83, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1934-7.  Google Scholar

show all references

References:
[1]

E. Bayer-Fluckiger, Ideal lattices, in A Panorama of Number Theory or the View from Baker's Garden, Cambridge Univ. Press, Cambridge, 2002,165-184. doi: 10.1017/CBO9780511542961.012.  Google Scholar

[2]

E. Bayer-Fluckiger, Lattices and number fields, in Algebraic Geometry: Hirzebruch 70, Contemp. Math., 241, Amer. Math. Soc., Providence, RI, 1999, 69–84. doi: 10.1090/conm/241/03628.  Google Scholar

[3]

E. Bayer-Fluckiger, Upper bounds for Euclidean minima of algebraic number fields, J. Number Theory, 121 (2006), 305-323.  doi: 10.1016/j.jnt.2006.03.002.  Google Scholar

[4]

E. Bayer-Fluckiger and G. Nebe, On the Euclidian minimum of some real number fields, J. Théor. Nombres Bordeaux, 17 (2005), 437–454. doi: 10.5802/jtnb.500.  Google Scholar

[5]

E. Bayer-Fluckiger, F. Oggier and E. Viterbo, New algebraic constructions of rotated $\mathbb{Z}^n$-lattice constellations for the Rayleigh fading channel, IEEE Trans. Inform. Theory, 50 (2004), 702–714. doi: 10.1109/TIT.2004.825045.  Google Scholar

[6]

E. Bayer-Fluckiger and I. Suarez, Ideal lattices over totally real number fields and Euclidean minima, Arch. Math. (Basel), 86 (2006), 217–225. doi: 10.1007/s00013-005-1469-9.  Google Scholar

[7]

K. Boullé and J. C. Belfiore, Modulation scheme design for the Rayleigh fading channel, Proc. Conf. Information Science and System, (1992), 288–293. Google Scholar

[8]

J. Boutros, E. Viterbo, C. Rastello and J.-C. Belfiore, Good lattice constellations for both Rayleigh fading and Gaussian channels, IEEE Trans. Inform. Theory, 42 (1996), 502–518. doi: 10.1109/18.485720.  Google Scholar

[9]

H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, Ann. of Math. (2), 170 (2009), 1003–1050. doi: 10.4007/annals.2009.170.1003.  Google Scholar

[10]

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Fundamental Principles of Mathematical Sciences, 290, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4757-2249-9.  Google Scholar

[11]

J. H. Conway and N. J. A. Sloane, The optimal isodual lattice quantizer in three dimensions, Adv. Math. Commun., 1 (2007), 257–260. doi: 10.3934/amc.2007.1.257.  Google Scholar

[12]

P. Elia, B. A. Sethuraman and P. V. Kumar, Perfect space-time codes for any number of antennas, IEEE Trans. Inform. Theory, 53 (2007), 3853–3868. doi: 10.1109/TIT.2007.907502.  Google Scholar

[13]

X. Hou and F. Oggier, Modular lattices from a variation of Construction A over number fields, Adv. Math. Commun., 11 (2017), 719–745. doi: 10.3934/amc.2017053.  Google Scholar

[14]

G. C. Jorge, A. A. de Andrade, S. I. R. Costa and J. E. Strapasson, Algebraic constructions of densest lattices, J. Algebra, 429 (2015), 218–235. doi: 10.1016/j.jalgebra.2014.12.044.  Google Scholar

[15]

G. C. Jorge and S. I. R. Costa, On rotated $D_n$-lattices constructed via totally real number fields, Arch. Math. (Basel), 100 (2013), 323–332. doi: 10.1007/s00013-013-0501-8.  Google Scholar

[16]

G. C. Jorge, A. J. Ferrari and S. I. R. Costa, Rotated $D_n$-lattices, J. Number Theory, 132 (2012), 2397–2406. doi: 10.1016/j.jnt.2012.05.002.  Google Scholar

[17]

D. Micciancio and S. Goldwasser, Complexity of Lattice Problems. A Cryptographic Perspective, The Kluwer International Series in Engineering and Computer Science, 671, Kluwer Academic Publishers, Boston, MA, 2002. doi: 10.1007/978-1-4615-0897-7.  Google Scholar

[18]

F. Oggier, Algebraic Methods for Channel Coding, Ph.D Thesis, École Polytechnique Fédérale in Lausanne, Lausanne, 2005. Google Scholar

[19]

F. Oggier and E. Bayer-Fluckiger, Best rotated cubic lattice constellations for the Rayleigh fading channel, Proceedings of IEEE International Symposium on Information Theory, Yokohama, Japan, 2003. Google Scholar

[20]

P. Samuel, Algebraic Theory of Numbers, Houghton Mifflin Co., Boston, MA, 1970,109pp.  Google Scholar

[21]

L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, 83, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1934-7.  Google Scholar

Table 1.  Normalized minimum product distance versus center density (from [5,12,15,16,18,19] and the results presented here)
$r $ $n $ $\sqrt[n]{d_{p}(\mathbb{Z}^n)} $ $\sqrt[n]{d_{p}(D_n)} $ $\sqrt[n]{d_{p}(A_n)} $ $\delta(\mathbb{Z}^n) $ $\delta(D_n) $ $\delta(A_n) $
$4 $ $3 $ $0.52275 $ $0.41491 $ $0.44544 $ $0.12500 $ $0.17677 $ $0.17677 $
$5 $ $7 $ $0.30080 $ $- $ $0.27602 $ $0.00780 $ $0.04419 $ $0.03125 $
$6 $ $15 $ $0.20138 $ $0.19229 $ $0.18513 $ $0.00003 $ $0.00276 $ $0.00138 $
$7 $ $31 $ $0.06220 $ $- $ $0.12782 $ $10^{-10} $ $10^{-5} $ $10^{-6} $
$8 $ $63 $ $0.09221 $ $0.09120 $ $0.08936 $ $10^{-19} $ $10^{-10} $ $10^{-11} $
$9 $ $127 $ $0.04542 $ $- $ $0.06284 $ $10^{-39} $ $10^{-20} $ $10^{-21} $
$10 $ $255 $ $0.03172 $ $- $ $0.04431 $ $10^{-77} $ $10^{-39} $ $10^{-40} $
$11 $ $511 $ $0.01819 $ $- $ $0.03129 $ $10^{-154} $ $10^{-78} $ $10^{-79} $
$12 $ $1023 $ $0.01569 $ $- $ $0.02211 $ $10^{-308} $ $10^{-155} $ $10^{-152} $
$13 $ $2047 $ $0.00522 $ $- $ $0.01563 $ $10^{-617} $ $10^{-309} $ $10^{-310} $
$14 $ $4095 $ $0.01106 $ $0.01106 $ $0.01106 $ $10^{-1233} $ $10^{-617} $ $10^{-619} $
$15 $ $8191 $ $0.00163 $ $- $ $0.00781 $ $10^{-2466} $ $10^{-1234} $ $10^{-1235} $
$16 $ $16383 $ $0.00319 $ $- $ $0.00552 $ $10^{-4932} $ $10^{-2467} $ $10^{-2468} $
$17 $ $32767 $ $0.00130 $ $- $ $0.00390 $ $10^{-9864} $ $10^{-4933} $ $10^{-4935} $
$18 $ $65535 $ $0.00276 $ $0.00276 $ $0.00276 $ $10^{-19729} $ $10^{-9865} $ $10^{-9867} $
$19 $ $131071 $ $0.00079 $ $- $ $0.00195 $ $10^{-39457} $ $10^{-19729} $ $10^{-19731} $
$20 $ $262143 $ $0.00138 $ $0.00138 $ $0.00138 $ $10^{-78913} $ $10^{-39457} $ $10^{-39460} $
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$r $ $n $ $\sqrt[n]{d_{p}(\mathbb{Z}^n)} $ $\sqrt[n]{d_{p}(D_n)} $ $\sqrt[n]{d_{p}(A_n)} $ $\delta(\mathbb{Z}^n) $ $\delta(D_n) $ $\delta(A_n) $
$4 $ $3 $ $0.52275 $ $0.41491 $ $0.44544 $ $0.12500 $ $0.17677 $ $0.17677 $
$5 $ $7 $ $0.30080 $ $- $ $0.27602 $ $0.00780 $ $0.04419 $ $0.03125 $
$6 $ $15 $ $0.20138 $ $0.19229 $ $0.18513 $ $0.00003 $ $0.00276 $ $0.00138 $
$7 $ $31 $ $0.06220 $ $- $ $0.12782 $ $10^{-10} $ $10^{-5} $ $10^{-6} $
$8 $ $63 $ $0.09221 $ $0.09120 $ $0.08936 $ $10^{-19} $ $10^{-10} $ $10^{-11} $
$9 $ $127 $ $0.04542 $ $- $ $0.06284 $ $10^{-39} $ $10^{-20} $ $10^{-21} $
$10 $ $255 $ $0.03172 $ $- $ $0.04431 $ $10^{-77} $ $10^{-39} $ $10^{-40} $
$11 $ $511 $ $0.01819 $ $- $ $0.03129 $ $10^{-154} $ $10^{-78} $ $10^{-79} $
$12 $ $1023 $ $0.01569 $ $- $ $0.02211 $ $10^{-308} $ $10^{-155} $ $10^{-152} $
$13 $ $2047 $ $0.00522 $ $- $ $0.01563 $ $10^{-617} $ $10^{-309} $ $10^{-310} $
$14 $ $4095 $ $0.01106 $ $0.01106 $ $0.01106 $ $10^{-1233} $ $10^{-617} $ $10^{-619} $
$15 $ $8191 $ $0.00163 $ $- $ $0.00781 $ $10^{-2466} $ $10^{-1234} $ $10^{-1235} $
$16 $ $16383 $ $0.00319 $ $- $ $0.00552 $ $10^{-4932} $ $10^{-2467} $ $10^{-2468} $
$17 $ $32767 $ $0.00130 $ $- $ $0.00390 $ $10^{-9864} $ $10^{-4933} $ $10^{-4935} $
$18 $ $65535 $ $0.00276 $ $0.00276 $ $0.00276 $ $10^{-19729} $ $10^{-9865} $ $10^{-9867} $
$19 $ $131071 $ $0.00079 $ $- $ $0.00195 $ $10^{-39457} $ $10^{-19729} $ $10^{-19731} $
$20 $ $262143 $ $0.00138 $ $0.00138 $ $0.00138 $ $10^{-78913} $ $10^{-39457} $ $10^{-39460} $
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