Papers by João Eloir Strapasson
It is shown that, given any k-dimensional lattice Λ, there is a lattice sequence Λ_w, w∈ Z, with ... more It is shown that, given any k-dimensional lattice Λ, there is a lattice sequence Λ_w, w∈ Z, with sub-orthogonal lattice Λ_o ⊂Λ, converging to Λ (unless equivalence), also we discuss the conditions for faster convergence.
We consider quasi-perfect codes in Z^n over the ℓ_p metric, 2 ≤ p < ∞. Through a computational... more We consider quasi-perfect codes in Z^n over the ℓ_p metric, 2 ≤ p < ∞. Through a computational approach, we determine all radii for which there are linear quasi-perfect codes for p = 2 and n = 2, 3. Moreover, we study codes with a certain degree of imperfection, a notion that generalizes the quasi-perfect codes. Numerical results concerning the codes with the smallest degree of imperfection are presented.
A method for finding an optimum n-dimensional commutative group code of a given order M is presen... more A method for finding an optimum n-dimensional commutative group code of a given order M is presented. The approach explores the structure of lattices related to these codes and provides a significant reduction in the number of non-isometric cases to be analyzed. The classical factorization of matrices into Hermite and Smith normal forms and also basis reduction of lattices are used to characterize isometric commutative group codes. Several examples of optimum commutative group codes are also presented.
This paper is a strongly geometrical approach to the Fisher distance, which is a measure of dissi... more This paper is a strongly geometrical approach to the Fisher distance, which is a measure of dissimilarity between two probability distribution functions. The Fisher distance, as well as other divergence measures, are also used in many applications to establish a proper data average. The main purpose is to widen the range of possible interpretations and relations of the Fisher distance and its associated geometry for the prospective applications. It focuses on statistical models of the normal probability distribution functions and takes advantage of the connection with the classical hyperbolic geometry to derive closed forms for the Fisher distance in several cases. Connections with the well-known Kullback-Leibler divergence measure are also devised.
Abstract Circulant graphs are homogeneous graphs with special properties which have been used to ... more Abstract Circulant graphs are homogeneous graphs with special properties which have been used to build interconnection networks for parallel computing. The association of circulant graphs to a spherical code in dimension 2k is presented here via the construction of an isomorphic graph supported by a lattice in R k.
Linear Algebra and its Applications, 2010
Circulant graphs are characterized here as quotient lattices, which are realized as vertices conn... more Circulant graphs are characterized here as quotient lattices, which are realized as vertices connected by a knot on a k-dimensional flat torus tessellated by hypercubes or hyperparallelotopes. Via this approach we present geometric interpretations for a bound on the diameter of a circulant graph, derive new bounds for the genus of a class of circulant graphs and establish connections with spherical codes and perfect codes in Lee spaces.
Designs, Codes and Cryptography, 2013
A method for finding an optimum n-dimensional commutative group code of a given order M is presen... more A method for finding an optimum n-dimensional commutative group code of a given order M is presented. The approach explores the structure of lattices related to these codes and provides a significant reduction in the number of non-isometric cases to be analyzed. The classical factorization of matrices into Hermite and Smith normal forms and also basis reduction of lattices are used to characterize isometric commutative group codes. Several examples of optimum commutative group codes are also presented.
In this paper we study sequences of lattices which are, up to similarity, projections of Z^n+1 on... more In this paper we study sequences of lattices which are, up to similarity, projections of Z^n+1 onto a hyperplane v^, with v∈Z^n+1 and converge to a target lattice Λ which is equivalent to an integer lattice. We show a sufficient condition to construct sequences converging at rate O(1/ |v|^2/n) and exhibit explicit constructions for some important families of lattices.
The genus graphs have been studied by many authors, but just a few results concerning in special ... more The genus graphs have been studied by many authors, but just a few results concerning in special cases: Planar, Toroidal, Complete, Bipartite and Cartesian Product of Bipartite. We present here a derive general lower bound for the genus of a abelian Cayley graph and construct a family of circulant graphs which reach this bound.
Resumo: O objetivo deste trabalhoé estudar ladrilhamentos de R n por poliominós associados a norm... more Resumo: O objetivo deste trabalhoé estudar ladrilhamentos de R n por poliominós associados a norma l p. No caso p = 1, a Conjectura de Golomb-Welch, em aberto por cerca de 44 anos, propõe que não existe ladrilhamento por tais poliominós, exceto em casos muito especiais. Aqui estamos interessados em discutir a existência de ladrilhamentos no caso geral 1 ≤ p ≤ ∞. Apresentamos resultados não-existenciais de ladrilhamentos para diferentes valores de p, bem como um estudo da forma e das propriedades combinatórias dos poliominós associados.
Discrete Applied Mathematics, 2014
This paper is a strongly geometrical approach to the Fisher distance, which is a measure of dissi... more This paper is a strongly geometrical approach to the Fisher distance, which is a measure of dissimilarity between two probability distribution functions. The Fisher distance, as well as other divergence measures, are also used in many applications to establish a proper data average. The main purpose is to widen the range of possible interpretations and relations of the Fisher distance and its associated geometry for the prospective applications. It focuses on statistical models of the normal probability distribution functions and takes advantage of the connection with the classical hyperbolic geometry to derive closed forms for the Fisher distance in several cases. Connections with the well-known Kullback-Leibler divergence measure are also devised.
Linear Algebra and Its Applications, 2013
In this paper we prove that given any two point lattices Λ 1 ⊂ R n and Λ 2 ⊂ R n-k , there is a s... more In this paper we prove that given any two point lattices Λ 1 ⊂ R n and Λ 2 ⊂ R n-k , there is a set of k vectors v i ∈ Λ 1 such that Λ 2 is, up to similarity, arbitrarily close to the projection of Λ 1 onto the orthogonal complement of the subspace spanned by v 1 , . . . , v k . This result extends the main theorem of [1] and has applications in communication theory.
2014 Information Theory and Applications Workshop (ITA), 2014
Codes and associated lattices are studied in the lp metric, particularly in the l1 (Lee) and the ... more Codes and associated lattices are studied in the lp metric, particularly in the l1 (Lee) and the l∞ (maximum) distances. Discussions and results on decoding processes, classification and analysis of perfect or dense codes in these metrics are presented.
Proceeding Series of the Brazilian Society of Computational and Applied Mathematics
Resumo. Este trabalho consiste numa investigação sobre a quantidade de Códigos Perfeitos, subreti... more Resumo. Este trabalho consiste numa investigação sobre a quantidade de Códigos Perfeitos, subreticulados, em Reticulados Algébricos bidimensionais obtidos via mergulho de Minkowski. Em contraste com os Códigos Perfeitos nos reticulados Z 2 e Hexagonal, considerar os Reticulados Algébricos resultou em um número grande de Códigos Perfeitos. Palavras-chave. Reticulados Algébricos, Código Perfeitos, Mergulho de Minkowski. 1 Introdução Ao longo dosúltimos anos, diversos autores exploraram a existência e a não existência de Códigos Perfeitos (no reticulado Z n), vide [4,6,7,9,10]. Em especial os autores Golomb and Welch, em [5], conjecturam que na métrica 1 (métrica do táxi) os perfeitos existiam para dimensões baixas e raios pequenos. E, e então diversos autores deram suas contribuições reforçando a conjectura. Constata-se que há poucos Códigos Perfeitos no ambiente Z n , quando se leva em conta as métricas: do táxi e a métrica euclidiana. Então outros autores começaram a explorar as métricas
Lattices are homogeneous discrete sets in the n-dimensional space that have been used in differen... more Lattices are homogeneous discrete sets in the n-dimensional space that have been used in different applications in communication areas such as coding for Gaussian or fading channels and cryptography. This chapter approaches the connection between quotient of lattices and spherical codes, presenting a survey on contributions to this topic mainly based on Costa et al. (Flat tori, lattices and spherical codes. In: 2013 Information Theory and Applications Workshop (ITA), February 2013, pp 1–8), Siqueira and Costa (Des Codes Cryptogr 49(1–3):307–321, 2008), Torezzan et al. (IEEE Trans Inf Theory 59(10):6655–6663, 2013), Costa et al. (Lattice applied to codding for reliable and secure communications. Springer, 2017), and Torezzan et al. (Des Codes Cryptogr 74(2):379–394, 2015).
ArXiv, 2016
It is shown that, given any $k$-dimensional lattice $\Lambda$, there is a lattice sequence $\Lamb... more It is shown that, given any $k$-dimensional lattice $\Lambda$, there is a lattice sequence $\Lambda_w$, $w\in \mathbb Z$, with suborthogonal lattice $\Lambda_o \subset \Lambda$, converging to $\Lambda$ (unless equivalence), also we discuss the conditions for faster convergence.
ArXiv, 2015
We consider quasi-perfect codes in Z over the `p metric, 2 ≤ p <∞. Through a computational app... more We consider quasi-perfect codes in Z over the `p metric, 2 ≤ p <∞. Through a computational approach, we determine all radii for which there are linear quasi-perfect codes for p = 2 and n = 2, 3. Moreover, we study codes with a certain degree of imperfection, a notion that generalizes the quasi-perfect codes. Numerical results concerning the codes with the smallest degree of imperfection are presented. ———————————————————————keywords:Tilings, Lattices, Quasi-perfect Codes, `p metric
Journal of Algebra and Its Applications
In this paper, we construct some families of rotated unimodular lattices and rotated direct sum o... more In this paper, we construct some families of rotated unimodular lattices and rotated direct sum of Barnes–Wall lattices [Formula: see text] for [Formula: see text] and [Formula: see text] via ideals of the ring of the integers [Formula: see text] for [Formula: see text] and [Formula: see text]. We also construct rotated [Formula: see text] and [Formula: see text]-lattices via [Formula: see text]-submodules of [Formula: see text]. Our focus is on totally real number fields since the associated lattices have full diversity and then may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. The minimum product distances of such constructions are also presented here.
International Transactions in Operational Research, 2015
In this paper we propose a heuristic technique for distributing points on the surface of a unit n... more In this paper we propose a heuristic technique for distributing points on the surface of a unit n-dimensional Euclidean sphere, generated as the orbit of a finite cyclic subgroup of orthogonal matrices, the so called cyclic group codes. Massive numerical experiments were done and many new cyclic group codes have been obtained in several dimensions at various rate. The obtained results assure that the heuristic approach have performance comparable to a brute-force search technique with the advantage of having low complexity, allowing for designing codes with a large number of points in higher dimensions.
Information geometry is approached here by considering the statistical model of multivariate norm... more Information geometry is approached here by considering the statistical model of multivariate normal distributions as a Riemannian manifold with the natural metric provided by the Fisher information matrix. Explicit forms for the Fisher-Rao distance associated to this metric and for the geodesics of general distribution models are usually very hard to determine. In the case of general multivariate normal distributions lower and upper bounds have been derived. We approach here some of these bounds and introduce a new one discussing their tightness in specific cases.
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Papers by João Eloir Strapasson