arXiv subjects
Publications and source records attributed to João E. Strapasson.
We consider quasi-perfect codes in $\mathbb{Z}^n$ over the $\ell_p$ metric, $2 \leq p < \infty$. 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 \textit{degree of imperfection}, a notion that generalizes the quasi-perfect codes. Numerical results concerning the codes with the smallest degree of imperfection are presented.
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.
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.
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.
The \emph{canonical structures of the plane} are those that result, up to isomorphism, from the rings that have the form $\mathds{R}[x]/(ax^2+bx+c)$ with $a\neq 0$.That ring is isomorphic to $\mathds{R}[θ]$, where $θ$ is the equivalence class of x, which satisfies $θ^2 = (-\dfrac{c}{a}) + θ(-\dfrac{b}{a})$. On the other hand, it is known that, up to isomorphism, there are only three canonical structures: the corresponding to $θ^2 = -1$ (the complex numbers), $θ^2 = 1$ (the perplex or hyperbolic numbers) and $θ^2 = 0$ (the parabolic numbers). This article copes with the algebraic structure of the rings of integers $\mathds{Z}[θ]$ in the perplex and parabolic cases by \emph{analogy} to the complex cases: the ring of Gaussian integers. For those rings a \emph{division algorithm} is proved and it is obtained, as a consequence, the characterization of the prime and irreducible elements.