Constructions of Dense Lattices of Full Diversity
DOI:
https://doi.org/10.5540/tema.2020.021.02.299Keywords:
Sphere packings, algebraic lattices, number fields, cyclotomic fields.Abstract
A lattice construction using Z-submodules of rings of integers of number fields is presented. The construction yields rotated versions of the laminated lattices A_n for n = 2,3,4,5,6, which are the densest lattices in their respective dimensions. The sphere packing density of a lattice is a function of its packing radius, which in turn can be directly calculated from the minimum squared Euclidean norm of the lattice. Norms in a lattice that is realized by a totally real number field can be calculated by the trace form of the field restricted to its ring of integers. Thus, in the present work, we also present the trace form of the maximal real subfield of a cyclotomic field. Our focus is on totally real number fields since their associated lattices have full diversity. Along with high packing density, the full diversity feature is desirable in lattices that are used for signal transmission over both Gaussian and Rayleigh fading channels.
References
A. A. Andrade and R. Palazzo Jr. Linear codes over finite rings, TEMA - Trends in Applied and Computational Mathematics, 6(2) (2005), 207-217.
A. S. Ansari, R. Shah, Zia Ur-Rahman, A. A. Andrade. Sequences of primitive and non-primitive BCH codes, TEMA - Trends in Applied and Computational Mathematics, 19(2) (2018), 369-389.
E. Bayer-Fluckiger. Lattices and number fields, In: Contemp. Math., Amer. Math. Soc., Providence (1999), 69-84.
E. Bayer-Fluckiger, F. Oggier and E. Viterbo. New algebraic constructions of rotated Z^n-lattice constellations for the Rayleigh fading channel, IEEE Trans. Inform. Theory, 50 (2004), 702-714.
J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups, 3rd Edition, Springer Verlag, New York (1999).
R. R. de Araujo, A. C. M. M. Chagas, A. A. Andrade and T. P. Nóbrega Neto. Trace form associated to cyclic number fields of ramified odd prime degree, accepted by J. Algebra App., June, 2019.
V. Baustista-Ancora and J. Uc-Kuk. The discriminant of abelian number fields, Journal of Mathematics, 47(1) (2017), 39-52.
E. Bayer-Fluckiger and I. S. Atias. Ideal lattices over totally real number fields and Euclidian minima, Archiv der Mathematik, 86(3) (2006) 217-225.
E. Bayer-Fluckiger and G. Nebe. On the Euclidian minimum of some real number fields, Journal de Théorie des Nombres de Bordeaux, 17(2) (2005) 437-454.
L. Washington. Introduction to cyclotomic fields, Springer-Verlag, New York (1995).
J. C. Interlando, T. P. Nóbrega Neto, T. M. Rodrigues and J. O. D. Lopes. A note on the integral trace form in cyclotomic fields, J. Algebra App., 14 (2015), 1550045
E. L. Oliveira, J. C. Interlando, T. P. da N'obrega Neto and J. O. D. Lopes. The integral trace form of cyclic extensions of odd prime degree, Rocky Mountain J. Math., 47 (2017), 1075-1088.
P. E. Conner and R. Perlis. A Survey of Trace Forms of Algebraic Number Fields, World Scientific Publishing Co Pte Ltd., Singapore (1984).
G. Lettl. The ring of integers of an Abelian number field, J. Reine Angew. Math., 404 (1990), 162-170.
Downloads
Published
How to Cite
Issue
Section
License
Copyright
Authors of articles published in the journal Trends in Computational and Applied Mathematics retain the copyright of their work. The journal uses Creative Commons Attribution (CC-BY) in published articles. The authors grant the TCAM journal the right to first publish the article.
Intellectual Property and Terms of Use
The content of the articles is the exclusive responsibility of the authors. The journal uses Creative Commons Attribution (CC-BY) in published articles. This license allows published articles to be reused without permission for any purpose as long as the original work is correctly cited.
The journal encourages Authors to self-archive their accepted manuscripts, publishing them on personal blogs, institutional repositories, and social media, as long as the full citation is included in the journal's website version.
