Constructions of algebraic lattices

In this work we present constructions of algebraic lattices in Euclidean space with optimal center density in dimensions 2, 3, 4, 6, 8 and 12, which are rotated versions of the lattices Λn, for n = 2,3,4,6,8 and K12. These algebraic lattices are constructed through twisted canonical homomorphism via ideals of a ring of algebraic integers. Mathematical subject classification: 18B35, 94A15, 20H10.

On the Optimality of Finite Constellations from Algebraic Lattices

A construction technique of finite point constellations in n-dimensional spaces from ideals in rings of algebraic integers is described. An algorithm is presented to find constellations with minimum average energy from a given lattice. For comparison, a numerical table of lattice constellations and group codes is computed for spaces of dimension two, three, and four. © 2001.

Constructions of ideal lattices with full diversity

In this paper, we present new constructions of ideal lattices for the Rayleigh fading channel in Euclidean spaces with full diversity. These constructions are through totally real subfields of cyclotomic fields, obtained by endowing their ring of integers. With this method we reproduce rotated versions of algebraic lattices where the performance in terms of minimum product distance is related with the field determinant.

Lattices from subfields of Q (xi n)

In this paper we present a method for evaluating the center density of algebraic lattices from subfields of Q(xi n), where n is a positive integer. This method allows to reproduce rotated versions of dense lattices in some dimensions. Constellations on algebraic lattices with high packing density have been proposed for use in communications in Gaussian channels and also in Rayleigh fading channels in case they have high diversity.

Construções de reticulados via extensões cíclicas de grau ímpar

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Corpos abelianos com aplicações

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Reticulados algébricos via corpos abelianos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Famílias de reticulados algébricos e reticulados ideais

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Discriminante de corpos de números

Pós-graduação em Matemática - IBILCE

Anéis de inteiros de corpos de números e aplicações

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Uma contribuição a teoria dos números e reticulados

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Algebraic constructions of densest lattices

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

An algebraic theory of functional and multivalued dependencies in relational databases

Computation of the dependency basis is the fundamental step in solving the membership problem for functional dependencies (FDs) and multivalued dependencies (MVDs) in relational database theory. We examine this problem from an algebraic perspective. We introduce the notion of the inference basis of a set M of MVDs and show that it contains the maximum information about the logical consequences of M. We propose the notion of a dependency-lattice and develop an algebraic characterization of inference basis using simple notions from lattice theory. We also establish several interesting properties of dependency-lattices related to the implication problem. Founded on our characterization, we synthesize efficient algorithms for (a): computing the inference basis of a given set M of MVDs; (b): computing the dependency basis of a given attribute set w.r.t. M; and (c): solving the membership problem for MVDs. We also show that our results naturally extend to incorporate FDs also in a way that enables the solution of the membership problem for both FDs and MVDs put together. We finally show that our algorithms are more efficient than existing ones, when used to solve what we term the ‘generalized membership problem’.

Lattices from maximal orders into quaternion algebras

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)