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Algebraic lattices via abelian number fields

Abstract

A lattice is a discrete additive subgroup of R^n. Signal constellations having lattice structure have been used as support for signal transmission over Gaussian and Rayleigh fading channels. Usually, the problem of finding good signal constellations for a Gaussian channel is associated to the search for latticeswith high packing density. On the other hand, for a Rayleigh fading channel the efficiency, measured by lower error probability in the transmission, is related to the lattice diversity and its minimum product distance. A lattice in R^n is called algebraic if it can be obtained as the image of a canonical or twisted embedding applied to a free Z-module of rank n contained in a number field of degree n. Algebraic lattice constructions may be used for calculating some lattice parameters, such as packing density and minimum product distance, which are difficult parameters to be calculated for general lattices in R^n. In this research project, making use of algebraic number theory, we propose the construction of families of lattices in R^n as algebraic lattices. It is of great interest to investigate in which number fields rotated versions of the lattices A_n, D_n, Z^n, E_6, E_7, E_8, K_ {12}, \Lambda_{16}, \Lambda_{24} and direct sums of them can be obtained via the canonical or twisted embeddings applied to fractional ideals. In some cases, our focus will be on totally real number fields since the lattices obtained therefrom are full diversity lattices. (AU)

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Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
STRAPASSON, JOAO E.; JORGE, GRASIELE C.; CAMPELLO, ANTONIO; COSTA, SUELI I. R.. Quasi-perfect codes in the l(p) metric. COMPUTATIONAL & APPLIED MATHEMATICS, v. 37, n. 2, p. 852-866, . (14/20602-8, 15/17167-0, 13/25977-7)

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