A Construction of Rotated Lattices via Totally Real Subfields of the Cyclotomic Field Q(z_p)
DOI:
https://doi.org/10.5540/tema.2019.020.03.561Keywords:
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.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).
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.
