E8-lattice via the cyclotomic field Q(ξ24)

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 E8 from the principal ideal I = ((1 + ξ3) + ξ3ξ24 + ξ3ξ24 2) of the cyclotomic field Q(ξ24), where ξ3 and ξ24 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.