E8-LATTICE VIA THE CYCLOTOMIC FIELD Q(ξ24)
C.C. Trinca Watanabe1, J.-C. Belfiore2,
E.D. De Carvalho3, J. Vieira Filho4
1Department of Communications (DECOM)
Campinas State University
Campinas - SP, 13083-852, BRAZIL
2Department of Communications and Electronics
Télécom ParisTech
Paris, 75013, FRANCE
3Department of Mathematics
São Paulo State University
Ilha Solteira - SP, 15385-000, BRAZIL
4Telecommunications Engineering
São Paulo State University
São João da Boa Vista - SP, 13876-750, BRAZIL

Abstract

Lattices can be applied in different areas of research, particularly, they can be applied in information theory and encryption schemes. Signal constellations having lattice structure have been used as a support for signal transmission over the Gaussian and Rayleigh fading channels.

The problem to find a good signal constellation for Gaussian channels is associated to the search of lattices which present a good packing density, that is, dense lattices. In this way, we propose an algebraic framework to construct the dense lattice $E_{8}$ from the principal ideal $\Im=((1+\xi_{3})+\xi_{3}\xi_{24}+\xi_{3}\xi_{24}^{2})$ of the cyclotomic field $\mathbb{Q}(\xi_{24})$, where $\xi_{3}$ and $\xi_{24}$ are the third and $24$-th root of unity, respectively.

The advantage of obtaining lattices from this method is the identification of the lattice points with the elements of a number field. Consequently, it is possible to utilize some properties of number fields in the study of such lattices.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 31
Issue: 1
Year: 2018

DOI: 10.12732/ijam.v31i1.6

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