A Construction of Rotated Lattices via Totally Real Subfields of the Cyclotomic Field Q(z_p)

Authors

  • Antonio A. Andrade São Paulo State University (Unesp), Institute of Biosciences, Humanities and Exact Sciences (Ibilce), Campus São José do Rio Preto.
  • Everton L. Oliveira Federal University of the Mato Grosso do Sul (UFMS), Campo Grande - MS.
  • José C. Interlando San Diego State University, San Diego, California.

DOI:

https://doi.org/10.5540/tema.2019.020.03.561

Keywords:

Lattices, cyclotomic fields, algebraic number field, rotated lattice.

Abstract

The theory of lattices have shown to be useful in information theory and rotated lattices with high modulations diversity have been extensively studied as an alternative approach for transmission over a Rayleigh-fading channel, where the performance of this modulation schemes essentially depends of the modulation diversity and of the minimum product distance to achieve substantial coding gains. The maximum diversity of a rotated lattice is guaranteed when we use totally real number fields and the minimum product distance is optimized by considering fields with minimum discriminant. In this paper, we present a construction of rotated lattice for the Rayleigh fading channel in Euclidean spaces with full diversity, where this construction is through a totally real subfield K of the cyclotomic field Q(z_p), where p is an odd prime, obtained by endowing their ring of integers.

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

Everton L. Oliveira, Federal University of the Mato Grosso do Sul (UFMS), Campo Grande - MS.

Department of Mathematics

José C. Interlando, San Diego State University, San Diego, California.

Department of Mathematics

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.

A. A. Andrade and J. C. Interlando. Rotated Z^n-lattices via real subfields of Q(z_{2^r}). TEMA - Thends in Applied and Computational Mathematics, to appear.

B. Erez. The Galois structure of the trace form in extensions of odd prime degree. J. Algebra, 118 (1988), 438-446.

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(4) (2004), 702-714.

J. Boutros, E. Viterbo, C. Rastello, and J.C. Belfiore. Good lattice constellations for both Rayleigh fading and Gaussian channels. IEEE Trans. Inf. Theory, 42(2) (1996), 502-518.

J. P. O. Santos. Introduction to numbers theory, Projeto Euclides, Impa, Rio de Janeiro (2006).

P. Elia, B. A. Sethuraman and P. Vijay Kumar. Perfect space-time codes for any number of antennas. IEEE Trans. Inform. Theory, 53(11) (2007), 3853-3868.

P. Ribenboin. Classical theory of algebraic numbers, Springer Verlag, New York (2001).

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Published

2019-12-02

How to Cite

Andrade, A. A., Oliveira, E. L., & Interlando, J. C. (2019). A Construction of Rotated Lattices via Totally Real Subfields of the Cyclotomic Field Q(z_p). Trends in Computational and Applied Mathematics, 20(3), 561. https://doi.org/10.5540/tema.2019.020.03.561

Issue

Vol. 20 No. 3 (2019)

Section

Original Article

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