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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