Ladrilhamentos reticulados de Z^n por esferas de Lee
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Data
2024-02-15
Autores
Ribeiro, Roberta Alves do Nascimento [UNIFESP]
Orientadores
Jorge, Grasiele Cristiane [UNIFESP]
Tipo
Dissertação de mestrado
Título da Revista
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Resumo
O objetivo deste trabalho é o estudo de ladrilhamentos reticulados de Z^n por esferas de Lee.
Investigaremos uma nova abordagem algébrica sobre esse problema, que é um caso especial da conjectura de Golomb–Welch. Utilizando esse novo método, é possível demonstrar a não existência de ladrilhamentos reticulados de Z^n por esferas de Lee com o mesmo raio r = 2 para infinitos valores da dimensão n. Tal método utiliza conceitos como os anéis de grupo e o grupo de caracteres, que conjuntamente oferecem um ambiente propício para uma nova abordagem utilizando um resultado conhecido acerca dos ladrilhamentos reticulados.
Neste estudo, damos ênfase a dois artigos: ``Perfect codes in the Lee metric and the packing of polyominoes'', de Solomon W. Golomb e Lloyd R. Welch, que apresenta a conjectura e enuncia alguns fatos envolvendo ladrilhamentos por esferas de Lee e ``On the nonexistence of lattice tilings of Z^nby Lee spheres'', de Tao Zhang e Yue Zhou, que soluciona alguns casos particulares da conjectura para r=2.
The aim of this work is to explore lattice tilings of Z^n by Lee spheres. We will examine a novel algebraic approach to this problem, a specific instance of the Golomb-Welch conjecture. Using this new method, it is possible to illustrate the absence of lattice tilings of Z^n by Lee spheres with the same radius r = 2 for infinitely many values of the dimension n. This method incorporates concepts such as group rings and character groups, providing a favorable environment for a fresh perspective utilizing a known result concerning lattice tilings. In this study, we highlight two articles: "Perfect codes in the Lee metric and the packing of polyominoes" by Solomon W. Golomb and Lloyd R. Welch, which introduces the conjecture and outlines some facts related to tilings by Lee spheres, and "On the nonexistence of lattice tilings of Z^n by Lee spheres" by Tao Zhang and Yue Zhou, which resolves certain specific cases of the conjecture for r=2.
The aim of this work is to explore lattice tilings of Z^n by Lee spheres. We will examine a novel algebraic approach to this problem, a specific instance of the Golomb-Welch conjecture. Using this new method, it is possible to illustrate the absence of lattice tilings of Z^n by Lee spheres with the same radius r = 2 for infinitely many values of the dimension n. This method incorporates concepts such as group rings and character groups, providing a favorable environment for a fresh perspective utilizing a known result concerning lattice tilings. In this study, we highlight two articles: "Perfect codes in the Lee metric and the packing of polyominoes" by Solomon W. Golomb and Lloyd R. Welch, which introduces the conjecture and outlines some facts related to tilings by Lee spheres, and "On the nonexistence of lattice tilings of Z^n by Lee spheres" by Tao Zhang and Yue Zhou, which resolves certain specific cases of the conjecture for r=2.