Constructions of Dense Lattices of Full Diversity

Authors

  • A. A. Andrade Departamento de Matemática, Ibilce - Unesp, São José do Rio Preto - SP
  • A. J. Ferrari Faculdade de Ciências, Unesp, Bauru - SP.
  • J. C. Interlando San Diego State University, San Diego, California, USA.
  • R. R. Araujo Universidade de São Paulo, São Paulo - SP.

DOI:

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

Keywords:

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.

Author Biographies

A. A. Andrade, Departamento de Matemática, Ibilce - Unesp, São José do Rio Preto - SP

Departamento de Matemática

A. J. Ferrari, Faculdade de Ciências, Unesp, Bauru - SP.

Departamento de Matemática

J. C. Interlando, San Diego State University, San Diego, California, USA.

Department of Mathematics & Statistics

R. R. Araujo, Universidade de São Paulo, São Paulo - SP.

Departamento de Matemática.

References

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Published

2020-07-22

How to Cite

Andrade, A. A., Ferrari, A. J., Interlando, J. C., & Araujo, R. R. (2020). Constructions of Dense Lattices of Full Diversity. Trends in Computational and Applied Mathematics, 21(2), 299. https://doi.org/10.5540/tema.2020.021.02.299

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Original Article