### Constructions of Dense Lattices of Full Diversity

#### 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.

#### Keywords

#### Full Text:

PDF#### 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.

DOI: https://doi.org/10.5540/tema.2020.021.02.299

#### Article Metrics

_{Metrics powered by PLOS ALM}

### Refbacks

- There are currently no refbacks.

**Trends in Computational and Applied Mathematics**

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

Indexed in: