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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

A brief survey of self-dual codes

Oktavia, Rini 2009 August 1900 (has links)
This report is a survey of self-dual binary codes. We present the fundamental MacWilliams identity and Gleason’s theorem on self-dual binary codes. We also examine the upper bound of minimum weights of self-dual binary codes using the extremal weight enumerator formula. We describe the shadow code of a self-dual code and the restrictions of the weight enumerator of the shadow code. Then using the restrictions, we calculate the weight enumerators of self-dual codes of length 38 and 40 and we obtain the known weight enumerators of this lengths. Finally, we investigate the Gaborit-Otmani experimental construction of selfdual binary codes. This construction involves a fixed orthogonal matrix, and we compare the result to the results obtained using other orthogonal matrices. / text
2

Identidades de MacWilliams para métricas Poset-Block / MacWilliams identity for Poset-Block metrics

Pinheiro, Jerry Anderson, 1985- 19 August 2018 (has links)
Orientador: Marcelo Firer / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T01:37:04Z (GMT). No. of bitstreams: 1 Pinheiro_JerryAnderson_M.pdf: 892573 bytes, checksum: 1be2db17f4d773a9b51d785a0fc55609 (MD5) Previous issue date: 2011 / Resumo: Em 1963, F. J. MacWilliams desenvolveu as chamadas identidades de MacWilliams, que estabelecem, em particular, relações entre a distribuição de pesos de códigos possuindo alta taxa de informação e códigos com baixa dimensão. Consideramos neste trabalho a família de métricas poset-block, uma pouco explorada generalização tanto das métricas de bloco quanto das métricas poset, e consequentemente da clássica métrica de Hamming. Efetuamos uma descrição detalhada dos espaços munidos com tais métricas com ênfase na teoria de códigos e em seguida tratamos do problema que surge naturalmente neste contexto: a caracterização dos espaços que admitem uma identidade do tipo MacWilliams, ou seja, a classificação das métricas que permitem relacionar unicamente o espectro de um código com o de seu dual. A principal técnica utilizada nesta classificação é a teoria de caracteres sobre corpos finitos, incluindo aí a transformada de Hadamard, a fórmula da soma discreta de Poisson e as relações de ortogonalidade existente entre caracteres. Tal técnica foi proposta inicialmente por F. J. MacWilliams e utilizada posteriormente por H. K. Kim e D. H. Oh na classificação das métricas poset que admitem identidades do tipo MacWilliams. Nosso principal objetivo é portanto classificar os espaços poset-block que admitem uma identidade do tipo MacWilliams. Como conseqüência desta classificação, através dos polinômios de Krawtchouk, obteremos expressões explícitas para estas identidades / Abstract: In 1963, F. J. MacWilliams developed the so-called MacWilliams identities, which establish, in particular, relations between the weight distribution of codes having high information rate and codes with low dimension. In this work we consider the family of poset-block metrics, a little explored generalization of both error-block and poset metrics, and hence also of the classic Hamming metric. We perform a detailed description of the spaces equipped with such metrics with emphasis in the coding theory and then we treat the problem that arises naturally in this context: the characterization of the poset-block metrics that admit a MacWilliams-type identity, in other words, the classification of metrics that allow to relate uniquely the spectrum of a code with the spectrum of its dual. The main technique used in this classification is the theory of characters over finite fields, including the Hadamard transform, the discrete Poisson summation formula and the orthogonality relations between characters. Such techniques were proposed initially by F. J. MacWilliams and used posteriorly by H. K. Kim and D. H. Oh in the classification of the metrics that admit a type of MacWilliams identity. Our main goal is therefore to classify the poset-block spaces that admit a MacWilliams type identity. As consequence of this classification, through the Krawtchouk polynomials, we will obtain explicit expressions for those identities / Mestrado / Matematica / Mestre em Matemática

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