18 research outputs found
Near MDS poset codes and distributions
We study -ary codes with distance defined by a partial order of the
coordinates of the codewords. Maximum Distance Separable (MDS) codes in the
poset metric have been studied in a number of earlier works. We consider codes
that are close to MDS codes by the value of their minimum distance. For such
codes, we determine their weight distribution, and in the particular case of
the "ordered metric" characterize distributions of points in the unit cube
defined by the codes. We also give some constructions of codes in the ordered
Hamming space.Comment: 13 pages, 1 figur
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
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