Códigos cíclicos lineares com dual complementar sobre anéis finitos de característica ímpar
In this work, we investigate cyclic codes in a finite non-chain ring $R=\mathbb{F}_q[x]/(x^2-1)$. Denoting by $v$ the class of $x$ in this quotient, we have $R=\mathbb{F}_q+v\mathbb{F}_q$, where $v^2 = 1$. We establish sufficient and necessary conditions for a code over $R$ to be considered a linear...
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| Formato: | tesis de maestría |
| Estado: | Versión publicada |
| Fecha de publicación: | 2023 |
| País: | Brasil |
| Recursos: | Universidade Federal de Uberlândia (UFU) |
| Repositorio: | Repositório Institucional da UFU |
| Idioma: | portugués |
| OAI Identifier: | oai:repositorio.ufu.br:123456789/39087 |
| Acesso em linha: | https://repositorio.ufu.br/handle/123456789/39087 http://doi.org/10.14393/ufu.di.2023.408 |
| Access Level: | acceso abierto |
| Palavra-chave: | Anel não encadeado Non-chain ring Código cíclico Cyclic code Código dual Dual code Código LCD LCD code Código reversível Reversible code Função de Gray Gray map CNPQ::CIENCIAS EXATAS E DA TERRA Matemática Códigos numéricos Álgebra linear Funções numéricas |
| Resumo: | In this work, we investigate cyclic codes in a finite non-chain ring $R=\mathbb{F}_q[x]/(x^2-1)$. Denoting by $v$ the class of $x$ in this quotient, we have $R=\mathbb{F}_q+v\mathbb{F}_q$, where $v^2 = 1$. We establish sufficient and necessary conditions for a code over $R$ to be considered a linear code with dual complement (LCD). Furthermore, we explore the properties of the dual code and its relationship with LCD codes. We demonstrate that the Gray map of an LCD code of length $n$ in $\mathbb{F}_q+v\mathbb{F}_q$ results in an LCD code of length $2n$ in $\mathbb{F}_q^{2n}$, and other properties of them |
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