# American Institute of Mathematical Sciences

August  2022, 16(3): 439-447. doi: 10.3934/amc.2020118

## Rotated $A_n$-lattice codes of full diversity

 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  August 2022 Early access  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.

Citation: Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $A_n$-lattice codes of full diversity. Advances in Mathematics of Communications, 2022, 16 (3) : 439-447. 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. [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. [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. [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. [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. [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. [7] K. Boullé and J. C. Belfiore, Modulation scheme design for the Rayleigh fading channel, Proc. Conf. Information Science and System, (1992), 288–293. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [18] F. Oggier, Algebraic Methods for Channel Coding, Ph.D Thesis, École Polytechnique Fédérale in Lausanne, Lausanne, 2005. [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. [20] P. Samuel, Algebraic Theory of Numbers, Houghton Mifflin Co., Boston, MA, 1970,109pp. [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.

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. [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. [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. [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. [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. [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. [7] K. Boullé and J. C. Belfiore, Modulation scheme design for the Rayleigh fading channel, Proc. Conf. Information Science and System, (1992), 288–293. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [18] F. Oggier, Algebraic Methods for Channel Coding, Ph.D Thesis, École Polytechnique Fédérale in Lausanne, Lausanne, 2005. [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. [20] P. Samuel, Algebraic Theory of Numbers, Houghton Mifflin Co., Boston, MA, 1970,109pp. [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.
Normalized minimum product distance versus center density (from [5,12,15,16,18,19] and the results presented here)
 $r$ $n$ $\sqrt[n]{d_{p}(\mathbb{Z}^n)}$ $\sqrt[n]{d_{p}(D_n)}$ $\sqrt[n]{d_{p}(A_n)}$ $\delta(\mathbb{Z}^n)$ $\delta(D_n)$ $\delta(A_n)$ $4$ $3$ $0.52275$ $0.41491$ $0.44544$ $0.12500$ $0.17677$ $0.17677$ $5$ $7$ $0.30080$ $-$ $0.27602$ $0.00780$ $0.04419$ $0.03125$ $6$ $15$ $0.20138$ $0.19229$ $0.18513$ $0.00003$ $0.00276$ $0.00138$ $7$ $31$ $0.06220$ $-$ $0.12782$ $10^{-10}$ $10^{-5}$ $10^{-6}$ $8$ $63$ $0.09221$ $0.09120$ $0.08936$ $10^{-19}$ $10^{-10}$ $10^{-11}$ $9$ $127$ $0.04542$ $-$ $0.06284$ $10^{-39}$ $10^{-20}$ $10^{-21}$ $10$ $255$ $0.03172$ $-$ $0.04431$ $10^{-77}$ $10^{-39}$ $10^{-40}$ $11$ $511$ $0.01819$ $-$ $0.03129$ $10^{-154}$ $10^{-78}$ $10^{-79}$ $12$ $1023$ $0.01569$ $-$ $0.02211$ $10^{-308}$ $10^{-155}$ $10^{-152}$ $13$ $2047$ $0.00522$ $-$ $0.01563$ $10^{-617}$ $10^{-309}$ $10^{-310}$ $14$ $4095$ $0.01106$ $0.01106$ $0.01106$ $10^{-1233}$ $10^{-617}$ $10^{-619}$ $15$ $8191$ $0.00163$ $-$ $0.00781$ $10^{-2466}$ $10^{-1234}$ $10^{-1235}$ $16$ $16383$ $0.00319$ $-$ $0.00552$ $10^{-4932}$ $10^{-2467}$ $10^{-2468}$ $17$ $32767$ $0.00130$ $-$ $0.00390$ $10^{-9864}$ $10^{-4933}$ $10^{-4935}$ $18$ $65535$ $0.00276$ $0.00276$ $0.00276$ $10^{-19729}$ $10^{-9865}$ $10^{-9867}$ $19$ $131071$ $0.00079$ $-$ $0.00195$ $10^{-39457}$ $10^{-19729}$ $10^{-19731}$ $20$ $262143$ $0.00138$ $0.00138$ $0.00138$ $10^{-78913}$ $10^{-39457}$ $10^{-39460}$
 $r$ $n$ $\sqrt[n]{d_{p}(\mathbb{Z}^n)}$ $\sqrt[n]{d_{p}(D_n)}$ $\sqrt[n]{d_{p}(A_n)}$ $\delta(\mathbb{Z}^n)$ $\delta(D_n)$ $\delta(A_n)$ $4$ $3$ $0.52275$ $0.41491$ $0.44544$ $0.12500$ $0.17677$ $0.17677$ $5$ $7$ $0.30080$ $-$ $0.27602$ $0.00780$ $0.04419$ $0.03125$ $6$ $15$ $0.20138$ $0.19229$ $0.18513$ $0.00003$ $0.00276$ $0.00138$ $7$ $31$ $0.06220$ $-$ $0.12782$ $10^{-10}$ $10^{-5}$ $10^{-6}$ $8$ $63$ $0.09221$ $0.09120$ $0.08936$ $10^{-19}$ $10^{-10}$ $10^{-11}$ $9$ $127$ $0.04542$ $-$ $0.06284$ $10^{-39}$ $10^{-20}$ $10^{-21}$ $10$ $255$ $0.03172$ $-$ $0.04431$ $10^{-77}$ $10^{-39}$ $10^{-40}$ $11$ $511$ $0.01819$ $-$ $0.03129$ $10^{-154}$ $10^{-78}$ $10^{-79}$ $12$ $1023$ $0.01569$ $-$ $0.02211$ $10^{-308}$ $10^{-155}$ $10^{-152}$ $13$ $2047$ $0.00522$ $-$ $0.01563$ $10^{-617}$ $10^{-309}$ $10^{-310}$ $14$ $4095$ $0.01106$ $0.01106$ $0.01106$ $10^{-1233}$ $10^{-617}$ $10^{-619}$ $15$ $8191$ $0.00163$ $-$ $0.00781$ $10^{-2466}$ $10^{-1234}$ $10^{-1235}$ $16$ $16383$ $0.00319$ $-$ $0.00552$ $10^{-4932}$ $10^{-2467}$ $10^{-2468}$ $17$ $32767$ $0.00130$ $-$ $0.00390$ $10^{-9864}$ $10^{-4933}$ $10^{-4935}$ $18$ $65535$ $0.00276$ $0.00276$ $0.00276$ $10^{-19729}$ $10^{-9865}$ $10^{-9867}$ $19$ $131071$ $0.00079$ $-$ $0.00195$ $10^{-39457}$ $10^{-19729}$ $10^{-19731}$ $20$ $262143$ $0.00138$ $0.00138$ $0.00138$ $10^{-78913}$ $10^{-39457}$ $10^{-39460}$
 [1] Daniele Bartoli, Adnen Sboui, Leo Storme. Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes. Advances in Mathematics of Communications, 2016, 10 (2) : 355-365. doi: 10.3934/amc.2016010 [2] Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1 [3] Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307 [4] Javier de la Cruz, Michael Kiermaier, Alfred Wassermann, Wolfgang Willems. Algebraic structures of MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 499-510. doi: 10.3934/amc.2016021 [5] Peter Haïssinsky, Kevin M. Pilgrim. An algebraic characterization of expanding Thurston maps. Journal of Modern Dynamics, 2012, 6 (4) : 451-476. doi: 10.3934/jmd.2012.6.451 [6] Aihua Li. An algebraic approach to building interpolating polynomial. Conference Publications, 2005, 2005 (Special) : 597-604. doi: 10.3934/proc.2005.2005.597 [7] Elisa Gorla, Felice Manganiello, Joachim Rosenthal. An algebraic approach for decoding spread codes. Advances in Mathematics of Communications, 2012, 6 (4) : 443-466. doi: 10.3934/amc.2012.6.443 [8] Sihem Mesnager, Gérard Cohen. Fast algebraic immunity of Boolean functions. Advances in Mathematics of Communications, 2017, 11 (2) : 373-377. doi: 10.3934/amc.2017031 [9] Doston Jumaniyozov, Ivan Kaygorodov, Abror Khudoyberdiyev. The algebraic classification of nilpotent commutative algebras. Electronic Research Archive, 2021, 29 (6) : 3909-3993. doi: 10.3934/era.2021068 [10] Sujay Jayakar, Robert S. Strichartz. Average number of lattice points in a disk. Communications on Pure and Applied Analysis, 2016, 15 (1) : 1-8. doi: 10.3934/cpaa.2016.15.1 [11] Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145. [12] Yingjie Bi, Siyu Liu, Yong Li. Periodic solutions of differential-algebraic equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1383-1395. doi: 10.3934/dcdsb.2019232 [13] Vu Hoang Linh, Volker Mehrmann. Spectral analysis for linear differential-algebraic equations. Conference Publications, 2011, 2011 (Special) : 991-1000. doi: 10.3934/proc.2011.2011.991 [14] Jaume Llibre, Claudia Valls. Algebraic limit cycles for quadratic polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2475-2485. doi: 10.3934/dcdsb.2018070 [15] L. Yu. Glebsky and E. I. Gordon. On approximation of locally compact groups by finite algebraic systems. Electronic Research Announcements, 2004, 10: 21-28. [16] Feng Rong. Non-algebraic attractors on $\mathbf{P}^k$. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 977-989. doi: 10.3934/dcds.2012.32.977 [17] Heide Gluesing-Luerssen, Uwe Helmke, José Ignacio Iglesias Curto. Algebraic decoding for doubly cyclic convolutional codes. Advances in Mathematics of Communications, 2010, 4 (1) : 83-99. doi: 10.3934/amc.2010.4.83 [18] Marco Calderini. A note on some algebraic trapdoors for block ciphers. Advances in Mathematics of Communications, 2018, 12 (3) : 515-524. doi: 10.3934/amc.2018030 [19] Geir Bogfjellmo. Algebraic structure of aromatic B-series. Journal of Computational Dynamics, 2019, 6 (2) : 199-222. doi: 10.3934/jcd.2019010 [20] B. Harbourne, P. Pokora, H. Tutaj-Gasińska. On integral Zariski decompositions of pseudoeffective divisors on algebraic surfaces. Electronic Research Announcements, 2015, 22: 103-108. doi: 10.3934/era.2015.22.103

2021 Impact Factor: 1.015

## Tools

Article outline

Figures and Tables