Perfect codes in the Lee and Chebyshev metrics and iterating Rédei functions

Orientadores: Sueli Irene Rodrigues Costa, Daniel Nelson Panario RodriguezTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O conteúdo desta tese insere-se dentro de duas áreas de pesquisa muito ativas: a teoria de códigos corretores de erros e sistemas dinâmicos sobre corpos finitos. Para abordar problemas em ambos os tópicos introduzimos um tipo de sequência finita que chamamos v-séries. No conjunto destas definimos uma métrica que induz uma estrutura de poset usada no estudo das possíveis estruturas de grupo abeliano representadas por códigos perfeitos na métrica de Chebyshev. Por outro lado, cada v-série é associada a uma árvore com raiz, a qual terá um papel importante em resultados relacionados à estrutura dinâmica de iterações de funções de Rédei. Na teoria de códigos corretores de erros, estudamos códigos perfeitos na métrica de Lee e na métrica de Chebyshev (correspondentes à métrica lp para p=1 e p=infinito respetivamente). Os principais resultados aqui estão relacionados com a descrição dos códigos q-ários n-dimensionais com raio de empacotamento e que sejam perfeitos nestas métricas, a obtenção de suas matrizes geradoras e a classificação destes, a menos de isometrias e a menos de isomorfismos. Varias construções de códigos perfeitos e famílias interessantes destes códigos com respeito à métrica de Chebyshev são apresentadas. Em sistemas dinâmicos sobre corpos finitos centramos nossa atenção em iterações de funções de Rédei, sendo o principal resultado um teorema estrutural para estas funções, o qual permite estender vários resultados sobre funções de Rédei. Este teorema pode também ser aplicado para outras classes de funções permitindo obter provas alternativas mais simples de alguns resultados conhecidos como o número de componentes conexas, o número de pontos periódicos e o valor esperado para o período e preperíodo da aplicação exponencial sobre corpos finitosAbstract: The content of this thesis is inserted in two very active research areas: the theory of error correcting codes and dynamical systems over finite fields. To approach problems in both topics we introduce a type of finite sequence called v-series. A metric is introduced in the set of such sequences inducing a poset structure used to determine all possible abelian group structures represented by perfect codes in the Chebyshev metric. Moreover, each v-serie is associated with a rooted tree, which has an important role in results related to the cycle structure of iterating Rédei functions. Regarding the theory of error correcting codes, we study perfect codes in the Lee metric and Chebyshev metric (corresponding to the lp metric for p=1 and p=infinity, respectively). The main results here are related to the description of n-dimensional q-ary codes with packing radius e which are perfect in these metrics, obtaining their generator matrices and their classification up to isometry and up to isomorphism. Several constructions of perfect codes in the Chebyshev metric are given and interesting families of such codes are presented. Regarding dynamical system over finite fields we focus on iterating Rédei functions, where our main result is a structural theorem, which allows us to extend several results on Rédei functions. The above theorem can also be applied to other maps, allowing simpler proofs of some known results related to the number of components, the number of periodic points and the expected value for the period and preperiod for iterating exponentiations over finite fieldsDoutoradoMatematica AplicadaDoutor em Matemática Aplicada2012/10600-2FAPESPCAPE

Study of imperfection degree in sub-lattices of the integer lattice

Orientador: João Eloir StrapassonDissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Um dos grandes problemas em aberto na matemática até os dias de hoje é a questão do empacotamento esférico. Para tentar resolver este problema, tem-se estudado alguns fatores importantes inerentes a isso. Nesse trabalho apresentamos uma breve introdução à teoria de reticulados e teoria de códigos, onde trataremos conceitos como densidade de empacotamento e de cobertura. O objetivo deste trabalho é o estudo da densidade de empacotamento e de cobertura em reticulados relativos à norma p. Neste estudo enfatizaremos o artigo Quasi-perfect codes in the lp metric de Strapasson et al. [13] onde é estabelecida a noção de perfeição e imperfeição de reticulados relativos à norma p, e é apresentado um algoritmo que busca por reticulados perfeitos e quase-perfeitosAbstract: One of the major unsolved problems in Mathematics until the present day is the sphere packing issue. To try addressing this problem, some key factors related to this issue have been studied. We present a brief introduction to lattice theory and coding theory in the present paper in which we deal with concepts such as packing and covering densities. The aim of the present work is the study of packing and covering densities on lattices related to p-metric. On this study we will highlight the article Quasi-perfect codes in the lp metric from Strapasson et al. [13] where the concept of perfection and imperfection of lattices related to p-metric is established, and an algorithm which seeks perfect and quasi-perfect lattices is presentedMestradoMatematica Aplicada e ComputacionalMestre em Matemática Aplicada e Computaciona

50 Years of the Golomb--Welch Conjecture

Since 1968, when the Golomb--Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb--Welch conjecture. Further, new results on Golomb--Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given.Comment: 28 pages, 2 figure

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

New Bounds for Permutation Codes in Ulam Metric

New bounds on the cardinality of permutation codes equipped with the Ulam distance are presented. First, an integer-programming upper bound is derived, which improves on the Singleton-type upper bound in the literature for some lengths. Second, several probabilistic lower bounds are developed, which improve on the known lower bounds for large minimum distances. The results of a computer search for permutation codes are also presented.Comment: To be presented at ISIT 2015, 5 page

Relativistic models of magnetars: Nonperturbative analytical approach

In the present paper we focus on building simple nonperturbative analytical relativistic models of magnetars. With this purpose in mind we first develop a method for generating exact interior solutions to the static and axisymmetric Einstein-Maxwell-hydrodynamic equations with anisotropic perfect fluid and with pure poloidal magnetic field. Then using an explicit exact solution we present a simple magnetar model and calculate some physically interesting quantities as the surface elipticity and the total energy of the magnetized star.Comment: 10 pages, LaTe