Error-block codes and poset metrics

Let P = ({1, 2,..., n}, <=) be a poset, let V-1, V-2,...,V-n, be a family of finite-dimensional spaces over a finite field F-q and let V = V-1 circle plus V-2 circle plus ... V-n. In this paper we endow V with a poset metric such that the P-weight is constant on the non-null vectors of a component V-i, extending both the poset metric introduced by Brualdi et al. and the metric for linear error-block codes introduced by Feng et al.. We classify all poset block structures which admit the extended binary Hamming code [8; 4; 4] to be a one-perfect poset block code, and present poset block structures that turn other extended Hamming codes and the extended Golay code [24; 12; 8] into perfect codes. We also give a complete description of the groups of linear isometrics of these metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset metric spaces. In particular, we obtain the group of linear isometrics of the error-block metric spaces.Let P = ({1, 2,..., n}, <=) be a poset, let V-1, V-2,...,V-n, be a family of finite-dimensional spaces over a finite field F-q and let V = V-1 circle plus V-2 circle plus ... V-n. In this paper we endow V with a poset metric such that the P-weight is cons2195111FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOsem informaçã

Duality for poset codes

Orientador: Marcelo FirerTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação CientificaResumo: Considerando uma generalização da métrica de Hamming, a métrica ponderada por uma ordem parcial, fazemos uma descrição sistemática para os espaços com a métrica ponderada, dando ênfase aos códigos poset e à hierarquia de pesos contextualizada nesse novo ambiente. Técnicas de multiconjunto, para códigos ponderados, são utilizadas para estender o Teorema da Dualidade de Wei, uma relação entre as hierarquias do código e do seu dual. Como consequência desta Dualidade estendemos certos resultados sobre a discrepância, códigos MDS e uma relação entre a condição cadeia do código e do seu dual.Abstract: Considering a generalization of the Hamming metric, the metric weighted by a partial order, we make a systematic description of the spaces with those metrics, emphasizing poset codes and the weight hierarchy of weights of those codes. Techniques of multiset, for weighted codes, are used to extend the Duality Theorem of Wei, a relationship between the hierarchy of a code and its dual. As a consequence of Duality we extend some results about the discrepancy, MDS codes and a relationship between a chain code and its dual.DoutoradoMatematicaDoutor em Matemátic

Classification of perfect linear codes with crown poset structure

Brualdi et al. introduced the concept of poset codes. In this paper, we consider the problem of classifying all perfect linear codes when the set of coordinate positions is endowed with crown poset structure. We derive a Ramanujan-Nagell type diophantine equation which is satisfied by parameters of perfect linear P-code. Solving this equation, we characterize parameters of one and two error correcting perfect linear P-codes. (C) 2002 Elsevier Science B.V. All rights reserved.X1111sciescopu

Groups Of Linear Isometries On Poset Structures

Let V be an n-dimensional vector space over a finite field Fq and P = { 1, 2, ..., n } a poset. We consider on V the poset-metric dP. In this paper, we give a complete description of groups of linear isometries of the metric space (V, dP), for any poset-metric dP. © 2007 Elsevier B.V. All rights reserved.3081841164123Ahn, J., Kim, H.K., Kim, J.S., Kim, M., Classification of perfect linear codes with crown poset structure (2003) Discrete Math., 268, pp. 21-30Brualdi, R., Graves, J.S., Lawrence, M., Codes with a poset metric (1995) Discrete Math., 147, pp. 57-72S.H. Cho, D.S. Kim, Automorphism group of the crown-weight space, submitted for publicationHyun, J.Y., Kim, H.K., The poset structures admitting the extended binary Hamming code to be a perfect code (2004) Discrete Math., 288, pp. 37-47Jang, Y., Park, J., On a MacWilliams type identity and a perfectness for a binary linear (n, n - 1, j)-poset code (2003) Discrete Math., 265, pp. 85-104D.S. Kim, Association schemes and MacWilliams dualities for generalized Rosenbloom-Tsfasman poset, submitted for publicationKim, D.S., Lee, J.G., A MacWilliams-type identity for linear codes on weak order (2003) Discrete Math., 262, pp. 181-194Lee, K., Automorphism group of the Rosenbloom-Tsfasman space (2003) European J. Combin., 24, pp. 607-612Lee, Y., Projective systems and perfect codes with a poset metric (2004) Finite Fields and Their Appl., 10, pp. 105-112Niederreiter, H., A combinatorial problem for vector spaces over finite fields (1991) Discrete Math., 96, pp. 221-228L. Panek, M. Firer, M. Muniz, Symmetry groups of Rosenbloom-Tsfasman spaces, submitted for publicationR. Stanley, Enumerative Combinatorics, vol. I, Wadsworth and Brooks/Cole, Monterey, CA, 1986Yu Rosenbloom, M., Tsfasman, M.A., Codes for the m-metric (1997) Problems Inform. Transmission, 33, pp. 45-5