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

Antonio A. Andrade, Everton L. Oliveira, José C. Interlando

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.

Keywords


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

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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).




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

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TEMA - Trends in Applied and Computational Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)
ISSN: 1677-1966  (print version),  2179-8451  (online version)

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