O segundo peso de Hamming do código de Reed-Muller generalizado

• Ano de defesa: 2016
• Autor(a) principal: Ávila, Dane Marques de
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal de Uberlândia; BR; Programa de Pós-graduação em Matemática; Ciências Exatas e da Terra; UFU
• Palavras-chave em Português: Códigos de Reed-Muller generalizados; Distância mínima; Segundo peso de Hamming; Códigos cartesianos Afins; Corpos finitos (Álgebra); Álgebra comutativa; Bases de Gröbner; Generalized Reed-Muller codes; Minimum distance; Second Hamming weight; Affine cartesian codes; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: https://repositorio.ufu.br/handle/123456789/16822; http://doi.org/10.14393/ufu.di.2016.166

In this work we present the determination of the second Hamming weight of generalized Reed- Muller codes in most cases (see Teorema 4.6). Our main reference is [13], although we have also used results from [3] and [5]. In the first chapter we describe finite fields e we show how they can be constructed. In chapter 2 we present the basics of coding theory. We define what are error correcting codes, the Hamming metric, the parameters of a code, the equivalence of codes through the concept of isometry, and we briefly present generalized Reed-Muller codes and their parameters. In chapter 3 we present some results from Grobner bases theory and the definition of Affine Cartesian codes, which generalize the generalized Reed-Muller codes. we use tools from Grobner bases theory to determine the dimension and the minimum distance of Affine Cartesian codes. We finish our work in chapter 4, with the determination of the second Hamming weight for generalized Reed-Muller codes in most cases.