The study of coding theory aims to detect and correct the errors during the transmission of the data. It enhances the quality of data transmission and provides better control over the noisy channels.The perfect codes are collected and analyzed in the premises of the Hamming metric.This classification yields that there exists only a few perfect codes. The perfect codes do not guarantee the perfection by all means but just satisfy certain bound and properties. The detection and correction of errors is always very important for better data transmission.

Este trabalho tem como objetivo discutir o conceito formal de distância em matemática, visando depois apresentar exemplos do conceito de distância em situações do dia a dia. Em geral com esse trabalho pretendemos que o leitor menos familiarizado entenda a importância do conceito matemático de distância. Distância é muito mais que o comprimento do segmento entre dois pontos e isso será apresentado em cada capítulo. O assunto foi inspirado pelo livro Encyclopedia of Distances Deza Michel Marie (2009), no qual são apresentados, espaços métricos, métricas em várias áreas e aplicações. No segundo capítulo, será apresentado a definição de espaços métricos. No terceiro capítulo serão apresentados alguns exemplos de métricas. As três primeiras métricas, as mais comuns: métricas euclidiana e máxima em R e R2. Também serão apresentadas as generalizações de cada uma delas em Rn. O próximo capítulo, o quarto, é destinado a apresentar o estudo sobre espaços normados, pois por meio desses conceitos pode-se analisar as distâncias entre vetores e matrizes. Veremos que a relevância dessas distâncias auxilia, por exemplo, na compreensão de aproximações de soluções de sistemas. No capítulo de distância de funções será apresentado um breve comentário sobre a série de Fourier, com relação ao método da aproximação através da decomposição de funções periódicas. Para analisar o quanto as funções trigonométricas estão se aproximando, usa-se o conceito de distância entre funções, as medições são feitas de acordo com as aproximações vão aumentando, essa distância \"erro\" entre elas tende a zero. Na teoria dos códigos, é preciso introduzir o conceito de distância entre \"palavras\", isso permite verificar se o código enviado teve alguma alteração, provocada por uma interferência ou ruídos durante a trajetória. Em algumas situações, o código consegue corrigir e compreender a palavra enviada mesmo tendo sofrido alterações no percurso. Nestes casos, há o estudo da métrica de Hamming. Já pela métrica de Hausdoorf, proposta pelo matemático de mesmo nome, é possível calcular com maior precisão a distância entre conjuntos fechados e limitados. Esta métrica pode ser utilizada em estudos de reconhecimento facial, por exemplo, pois as imagens das faces são transformadas em nuvens de pontos. Além disso, através do algoritmo de Dijkstra será apresentado a distância entre os vértices de um grafo convexo. Existem várias aplicações de distância entre grafos e uma delas é a questão de minimizar o custo decorrente do deslocamento entre uma transportadora e o local de entrega por exemplo. Para finalizar à discussão da importância do consenso de distância, será apresentada uma distância entre genes. Dentro deste tema, o principal cientista foi Thomas Morgan, que por meio de seus estudos conseguiu criar o primeiro mapeamento genético. Com isto, pode relacionar o conceito de distância entre genes à taxa de recombinação gênica. Finalmente, foi elaborada uma atividade com alunos do ensino médio com o objetivo de analisar os conhecimentos que os estudantes têm sobre distância. Esta atividade também foi importante para que os alunos pudessem compreender a necessidade de formalizar matematicamente este conceito e, principalmente, motivá-los por meio da apresentação de aplicações sobre distância, em diferentes âmbitos. / This work has as objective to discuss the formal concept of distance in mathematics, aiming to present examples of the distance concept in everyday situations. In general with this work we want the less familiar reader to understand the importance of the mathematical concept of distance. Distance is much more than the length of the segment between two points and this will be presented in each chapter. The subject was inspired by the book Encyclopedia of Distances Deza Michel Marie (2009), in which are presented, metric spaces, metrics in different areas and applications. In the second chapter, the definition of metric spaces will be presented. In the third chapter some examples of metrics will be presented. The first three metrics, the most common: usual, Euclidean, and maximum metrics in R and R2. Also the generalizations of each of them were presented in Rn. The next chapter, the fourth, is intended to show the study on normed spaces, because through these concepts we can analyze the distances between vectors and matrices. We will see that the relevance of these distances helps in the understanding of systems solutions approximation. In the chapter on distance of functions, a brief comment about Fourier series was presented, regarding the method of approximation through the decomposition of periodic functions. In order to analyze how the trigonometric functions are approaching, the concept of distance between functions is used, the measurements are made as the approximations increase, this distance \"error\" between them tends to zero. In codes theory, it is necessary to introduce the concept of distance between \"words\", this allows to verify if the code had some alteration, caused by an interference or noises during the trajectory. In some situations, the code can correct and understand the sent word even though it has undergone changes in the route. In these cases, there is Hammings metrics study. By the Hausdoorf metric, proposed by the mathematician of the same name, it is possible to calculate with more precision the distance between closed and limited sets. This metric can be used in face recognition studies, for example, because face images are transformed into clouds of dots. Then, through the Dijkstras algorithm will be presented the distance between the vertices of a convex graphic. There are several applications of distance between graphics and one of them is the issue of minimizing the cost of moving between a local carrier company and the place of delivery, for example. To finish the discussion about the importance of distance consensus, the distance between genes will be presented. Within this theme, the main scientist was Thomas Morgan, who through his studies managed to create the first genetic mapping. With this, he was able to relate the concept of distance between genes to the rate of gene recombination. Finally, an activity was elaborated with high school students with the objective of analyzing students knowledge about distance. This activity was also important so that the students could understand about a necessity to formalize this concept mathematically and, mainly, to motivate them through the presentation of applications on distance, in different scopes.

Orientador: Marcelo Firer / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T01:48:53Z (GMT). No. of bitstreams: 1 Moura_AllandeOliveira1_D.pdf: 766044 bytes, checksum: e752134a3aa77aa9bf3559d18a7f0a12 (MD5) Previous issue date: 2010 / Resumo: Considerando uma generalização da métrica de Hamming, a métrica ponderada por uma ordem parcial, fazemos uma descrição sistemática para os espaços com a métrica ponderada, dando ênfase aos códigos poset e à hierarquia de pesos contextualizada nesse novo ambiente. Técnicas de multiconjunto, para códigos ponderados, são utilizadas para estender o Teorema da Dualidade de Wei, uma relação entre as hierarquias do código e do seu dual. Como consequência desta Dualidade estendemos certos resultados sobre a discrepância, códigos MDS e uma relação entre a condição cadeia do código e do seu dual. / Abstract: Considering a generalization of the Hamming metric, the metric weighted by a partial order, we make a systematic description of the spaces with those metrics, emphasizing poset codes and the weight hierarchy of weights of those codes. Techniques of multiset, for weighted codes, are used to extend the Duality Theorem of Wei, a relationship between the hierarchy of a code and its dual. As a consequence of Duality we extend some results about the discrepancy, MDS codes and a relationship between a chain code and its dual. / Doutorado / Matematica / Doutor em Matemática

<p>Perfect codes and minimal distance of a code have great importance in the study of theoryof codes. The perfect codes are classified generally and in particular for the Lee metric.However, there are very few perfect codes in the Lee metric. The Lee metric hasnice properties because of its definition over the ring of integers residue modulo q. It isconjectured that there are no perfect codes in this metric for q > 3, where q is a primenumber.The minimal distance comes into play when it comes to detection and correction oferror patterns in a code. A few bounds on the number of codewords and minimal distanceof a code are discussed. Some examples for the codes are constructed and their minimaldistance is calculated. The bounds are illustrated with the help of the results obtained.</p>

The study of multicomputer interconnection networks is an important area of research in parallel processing. We introduce vertex-symmetric Hamming-group graphs as a model to design a wide variety of network topologies including the hypercube network.

Perfect codes and minimal distance of a code have great importance in the study of theoryof codes. The perfect codes are classified generally and in particular for the Lee metric.However, there are very few perfect codes in the Lee metric. The Lee metric hasnice properties because of its definition over the ring of integers residue modulo q. It isconjectured that there are no perfect codes in this metric for q &gt; 3, where q is a primenumber.The minimal distance comes into play when it comes to detection and correction oferror patterns in a code. A few bounds on the number of codewords and minimal distanceof a code are discussed. Some examples for the codes are constructed and their minimaldistance is calculated. The bounds are illustrated with the help of the results obtained.

A distributed hash table (DHT) is a type of peer-to-peer (P2P) network that, like traditional hash tables, maps keys to values. Unlike traditional hash tables, however, the data is distributed across a network with each node being responsible for a particular range of keys. Numerous other DHTs have been presented and have become the cornerstone of wildly popular P2P file-sharing applications, such as BitTorrent. Each of these DHTs trades-off the number of pointers maintained per node with the overhead and lookup time; storing more pointers decreases the lookup time at the expense of increased overhead. A Grouped Hamming Network (GHN), the overlay network presented in this thesis, allows for the number of pointers per node to be any increasing function of n, P(n) = Ω(log n). The system presented assumes that nodes fail independently and uniformly at random with some probability q = 1 − p. Three different schemes for routing in a GHN are presented. For each routing scheme a theoretical estimate on the probability of failure is given and optimal configurations in terms of n and P(n) are given. Simulations of GHNs with various configurations indicate that the given estimates are indeed accurate for reasonable values of q and that the optimal configurations are accurate.

A distributed hash table (DHT) is a type of peer-to-peer (P2P) network that, like traditional hash tables, maps keys to values. Unlike traditional hash tables, however, the data is distributed across a network with each node being responsible for a particular range of keys. Numerous other DHTs have been presented and have become the cornerstone of wildly popular P2P file-sharing applications, such as BitTorrent. Each of these DHTs trades-off the number of pointers maintained per node with the overhead and lookup time; storing more pointers decreases the lookup time at the expense of increased overhead. A Grouped Hamming Network (GHN), the overlay network presented in this thesis, allows for the number of pointers per node to be any increasing function of n, P(n) = Ω(log n). The system presented assumes that nodes fail independently and uniformly at random with some probability q = 1 − p. Three different schemes for routing in a GHN are presented. For each routing scheme a theoretical estimate on the probability of failure is given and optimal configurations in terms of n and P(n) are given. Simulations of GHNs with various configurations indicate that the given estimates are indeed accurate for reasonable values of q and that the optimal configurations are accurate.

This thesis evaluates various techniques for intra-set clustering of social media data from an industry perspective. The research goal was to establish methods for reducing the amount of redundant information an end user must review from a standard social media search. The research evaluated both clustering algorithms and string similarity measures for their effectiveness in clustering a selection of real-world topic and location-based social media searches. In addition, the algorithms and similarity measures were tested in scenarios based on industry constraints such as rate limits. The results were evaluated using several practical measures to determine which techniques were effective. / Graduate

abstract: In software testing, components are tested individually to make sure each performs as expected. The next step is to confirm that two or more components are able to work together. This stage of testing is often difficult because there can be numerous configurations between just two components. Covering arrays are one way to ensure a set of tests will cover every possible configuration at least once. However, on systems with many settings, it is computationally intensive to run every possible test. Test prioritization methods can identify tests of greater importance. This concept of test prioritization can help determine which tests can be removed with minimal impact to the overall testing of the system. This thesis presents three algorithms that generate covering arrays that test the interaction of every two components at least twice. These algorithms extend the functionality of an established greedy test prioritization method to ensure important components are selected in earlier tests. The algorithms are tested on various inputs and the results reveal that on average, the resulting covering arrays are two-fifths to one-half times smaller than a covering array generated through brute force. / Dissertation/Thesis / Masters Thesis Computer Science 2015