Trace forms of certain subfields of cyclotomic fields and applications

Year 2020, , 141 - 160, 07.05.2020

Agnaldo José Ferrarı Antonio Aparecıdo De Andrade Robson Rıcardo De Araujo José Carmelo Interlando

https://doi.org/10.13069/jacodesmath.729440

Abstract

In this work, we present a explicit trace forms for maximal real subfields of cyclotomic fields as tools for constructing algebraic lattices in Euclidean space with optimal center density. We also obtain a closed formula for the Gram matrix of algebraic lattices obtained from these subfields. The obtained lattices are rotated versions of the lattices $ \Lambda_9, \Lambda_{10}$ and $\Lambda_{11}$ and they are images of $\mathbb{Z}$-submodules of rings of integers under the twisted homomorphism, and these constructions, as algebraic lattices, are new in the literature. We also obtain algebraic lattices in odd dimensions up to $7$ over real subfields, calculate their minimum product distance and compare with those known in literatura, since lattices constructed over real subfields have full diversity.

Keywords

Cyclotomic fields, Algebraic lattices, Twisted homomorphism, Signal design

Thanks

This work was supported by Fapesp 2013/25977-7 and CNPq 429346/2018-2.

References

• [1] A. A. Andrade, A. J. Ferrari, C. W. O. Benedito, Constructions of algebraic lattices, Comput. Appl. Math. 29(3) (2010) 1–13.
• [2] E. Bayer–Fluckiger, Ideal lattices, Proceedings of the conference Number Theory and Diophantine Geometry (2002) 168–184.
• [3] E. Bayer–Fluckiger, Lattices and number fields, Contemp. Math. 241 (1999) 69–84.
• [4] E. Bayer–Fluckiger, Upper bounds for Euclidean minima of algebraic number fields, J. Number Theory 121(2) (2006) 305–323.
• [5] E. Bayer–Fluckiger, F. Oggier, E. Viterbo, New algebraic constructions of rotated $\mathbb{Z}^n$–lattice constellations for the Rayleigh fading channel, IEEE Trans. Inform. Theory 50(4) (2004) 702–714.
• [6] E. Bayer–Fluckiger, G. Nebe, On the Euclidian minimum of some real number fields, Journal de Théorie des Nombres de Bordeaux 17(2) (2005) 437–454.
• [7] E. Bayer–Fluckiger, I. Suarez, Ideal lattices over totally real number fields and Euclidean minima, Arch. Math. 86 (2006) 217–225.
• [8] E. Bayer–Fluckiger, P. Maciak, Upper bounds for Euclidean minimal for abelian number fields of odd prime conductor, Math. Ann. 357 (2013) 1071–1089.
• [9] W. Bosma, J. Cannon, C. Playoust, The magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997) 235–265.
• [10] J. Boutros, E. Viterbo, C. Rastello, J. C. Belfiori, Good lattice constellations for both Rayleigh fading and Gaussian channels, IEEE Trans. Inform. Theory 42(2) (1996) 502–518.
• [11] H. Cohn, A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, Ann. of Math. 170(3) (2009) 1003–1050.
• [12] J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer–Verlag, New York 1998.
• [13] A. J. Ferrari, A. A. Andrade, Constructions of rotated lattice constellations in dimensions power of 3, J. Algebra Appl. 17(09) (2017) 1850175.
• [14] J. C. Interlando, T. P. N. Neto, T. M. Rodrigues, J. O. D. Lopes, A note on the integral trace form in cyclotomic fields, J. Algebra Appl. 14(04) (2015) 1550045.
• [15] G. C. Jorge, A. A. Andrade, S. I. R. Costa, J. E. Strapasson, Algebraic constructions of densest lattices, J. Algebra 429 (2015) 218–235.
• [16] G. C. Jorge, A. J. Ferrari, S. I. R. Costa, Rotated $D_n$–lattices, J. Number Theory 132(11) (2012) 2397–2406.
• [17] D. Micciancio, S. Goldwasser, Complexity of Lattice Problems: A Cryptographic Perspective, in Kluwer Internat. Ser. Engrg. Comput. Sci., Kluwer Academic Publishers 671 2002.
• [18] F. Oggier, E. Bayer–Fluckiger, Best rotated cubic lattice constellations for the Rayleigh fading channel, in Proceedings of IEEE International Symposium on Information Theory (2003).
• [19] E. L. Oliveira, J. C. Interlando, T. P. N. Neto, J. O. D. Lopes, The integral trace form of cyclic extensions of odd prime degree, Rocky Mountain J. Math. 47(4) (2017) 1075–1088.
• [20] P. Samuel, Algebraic Theory of Numbers, Hermann, 1982.
• [21] L. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, 1995.