Rotated Z^n-Lattices via Real Subfields of Q(\zeta_2r)

Authors

  • Antonio A. Andrade São Paulo State University (Unesp), Institute of Biosciences, Humanities and Exact Sciences (Ibilce), Campus São José do Rio Preto. https://orcid.org/0000-0001-6452-2236
  • José C. Interlando San Diego State University

DOI:

https://doi.org/10.5540/tema.2019.020.03.445

Keywords:

Lattices, cyclotomic fields, modulation design, fading channels, minimum product distance.

Abstract

A method for constructing rotated Z^n-lattices, with n a power of 2, based on totally real subfields of the cyclotomic field Q(\zeta_{2^r}), where r\geq 4 is an integer, is presented. Lattices exhibiting full diversity in some dimensions n not previously addressed are obtained.

Author Biographies

Antonio A. Andrade, São Paulo State University (Unesp), Institute of Biosciences, Humanities and Exact Sciences (Ibilce), Campus São José do Rio Preto.

Department of Mathematics

José C. Interlando, San Diego State University

Department of Mathematics & Statistics

References

A. A. Andrade, T. Shah & A. Khan. A note on linear codes over semigroup rings. TEMA - Thends in Applied and Computational Mathematics, 12(2) (2011), 79-89.

A. Ansari, T. Shah, Zia Ur-Rahman & A. A. Andrade. Sequences of primitive and non-primitive BCH codes. TEMA - Thends in Applied and Computational Mathematics, 19(2) (2018), 369-389.

A. A. Andrade, C. Alves & T. B. Carlos, Rotated lattices via the cyclotomic field Q(zeta_{2^r}), Internat. J. Appl. Math., 19, 2006, 321-331.

V. Bautista-Ancora & J. Uc-Kuk, The discriminant of Abelian number fields, Rocky Mountain J. Math., 47, 2017, 39-52.

E. Bayer-Fluckiger, Lattices and number fields, Algebraic geometry: Hirzebruch 70, Contemp. Math., 241, Amer. Math. Soc., Providence, RI, 1999, 69-84.

E. Bayer-Fluckiger, F. Oggier & E. Viterbo. New algebraic constructions of rotated Z^n-lattice constellations for the Rayleigh fading channel. IEEE Trans. Inform. Theory, 50 (2004), 702-714.

E. Bayer-Fluckiger, F. Oggier & E. Viterbo. Algebraic lattice constellations: bounds on performance, IEEE Trans. Inform. Theory, 52 (2006), 319-327.

J. Boutros, E. Viterbo, C. Rastello & J. C. Belfiore. Good lattice constellations for both Rayleigh fading and Gaussian channels. IEEE Trans. Inform. Theory, 42 (1996), 502-518.

M. O. Damen, K. Abed-Meraim & J. C. Belfiori. Diagonal algebraic space-time block codes. IEEE Trans. Inform. Theory, 48 (2002), 628-636.

P. Elia, B. A. Sethuraman & P. V. Kumar. Perfect space-time codes for any number of antennas. IEEE Trans. Inform. Theory, 53 (2007), 3853-3868.

S. Lang. Algebra, revised third edition, Springer-Verlag, New York (2003).

A. M. Odlyzko. Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results. J.

Théorie des Nombres de Bordeaux 2 (1990), 119-141.

K. H. Rosen. Elementary Number Theory and its Applications, sixth edition, Addison-Wesley, Reading, MA (2011).

L. Washington. Introduction to Cyclotomic Fields, second edition, Springer-Verlag, New York (1997).

Downloads

Published

2019-12-02

How to Cite

Andrade, A. A., & Interlando, J. C. (2019). Rotated Z^n-Lattices via Real Subfields of Q(\zeta_2r). Trends in Computational and Applied Mathematics, 20(3), 445. https://doi.org/10.5540/tema.2019.020.03.445

Issue

Section

Original Article