Classification of poset-block spaces admitting MacWilliams-type identity

In this work we prove that a poset-block space admits a MacWilliams-type identity if and only if the poset is hierarchical and at any level of the poset, all the blocks have the same dimension. When the poset-block admits the MacWilliams-type identity we explicit the relation between the weight enumerators of a code and its dual.Comment: 8 pages, 1 figure. Submitted to IEEE Transactions on Information Theor

Bounds for complexity of syndrome decoding for poset metrics

In this work we show how to decompose a linear code relatively to any given poset metric. We prove that the complexity of syndrome decoding is determined by a maximal (primary) such decomposition and then show that a refinement of a partial order leads to a refinement of the primary decomposition. Using this and considering already known results about hierarchical posets, we can establish upper and lower bounds for the complexity of syndrome decoding relatively to a poset metric.Comment: Submitted to ITW 201

Coding and Decoding Schemes for MSE and Image Transmission

In this work we explore possibilities for coding and decoding tailor-made for mean squared error evaluation of error in contexts such as image transmission. To do so, we introduce a loss function that expresses the overall performance of a coding and decoding scheme for discrete channels and that exchanges the usual goal of minimizing the error probability to that of minimizing the expected loss. In this environment we explore the possibilities of using ordered decoders to create a message-wise unequal error protection (UEP), where the most valuable information is protected by placing in its proximity information words that differ by a small valued error. We give explicit examples, using scale-of-gray images, including small-scale performance analysis and visual simulations for the BSMC.Comment: Submitted to IEEE Transactions on Information Theor

Canonical form for poset codes and coding-decoding schemes for expected loss

Orientador: Marcelo FirerTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: No contexto de códigos corretores de erros, métricas são utilizadas para definir decodificadores de máxima proximidade, uma alternativa aos decodificadores de máxima verossimilhança. A família de métricas poset tem sido extensivamente estudada no contexto de teoria de códigos. Considerando a estrutura do grupo de isometrias lineares, é obtida uma forma canônica para matrizes geradoras de códigos lineares. Esta forma canônica permite obter expressões e limitantes analíticos para alguns invariantes clássicos da teoria: raio de empacotamento e complexidade de síndrome. Ainda, substituindo a probabilidade de erro pela perda esperada definida pelo desvio médio quadrático (entre a informação original e a informação decodificada), definimos uma proposta de codificação com ordem lexicográfica que, em algumas situações é ótima e em outras, as simulações feitas sugerem um desempenho ao menos subótimo. Finalmente, relacionamos a medida de perda esperada com proteção desigual de erros, fornecendo uma construção de códigos com dois níveis de proteção desigual de erros e com perda esperada menor que a obtida pelo produto de dois códigos ótimos, que separam as informações que são protegidas de modo diferenciadoAbstract: In the context of error-correcting codes, metrics are used to define minimum distance decoders, an alternative to maximum likelihood decoders. The family of poset metrics has been extensively studied in the context of coding theory. Considering the structure of the group of linear isometries, we obtain a canonical form for generator matrices of linear codes. The canonical form allows to obtain analytics expressions and bounds for classical invariants of the theory: packing radius and syndrome complexity. By substituting the error probability by the expected loss defined by the mean square deviation (between the original information and the decoded information), we propose an encoder scheme which, in some situations is optimal, and in others the simulations suggest a performance at least sub-optimal. Finally, we relate the expected loss measure with unequal error protection, providing a construction of codes with two levels of unequal error protection and expected loss smaller than the one obtained by the product of two optimal codes, which divide the information that is protected differentlyDoutoradoMatematicaDoutor em Matemática141586/2014-1CNPQCAPE

Combinatorial metrics : MacWilliams-type identities, isometries and extension property

In this work we characterize the combinatorial metrics admitting a MacWilliams-type identity and describe the group of linear isometries of such metrics. Considering the binary case, we classify the metrics satisfying the MacWilliams extension property (for disconnected coverings) and give a necessary condition for the extension property (for connected coverings)872-3327340FAPESP – Fundação de Amparo à Pesquisa Do Estado De São Paulo2013/25977-7; 2017/14616-4; 2017/10018-