Codes with a poset metric

AbstractNiederreiter generalized the following classical problem of coding theory: given a finite field Fq and integers n > k ⩾ 1, find the largest minimum distance achievable by a linear code over Fq of length n and dimension k. In this paper we place this problem in the more general setting of a partially ordered set and define what we call poset-codes. In this context, Niederreiter's setting may be viewed as the disjoint union of chains. We extend some of Niederreiter's bounds and also obtain bounds for posets which are the product of two chains

Symmetry groups of Rosenbloom-Tsfasman spaces

Let F(q)(m.n) he the vector space of m . n-tuples over a finite field Fit and P = {1, 2,..., m - n} a poset that is the finite union of In disjoint chains of length it. We consider on F(q)(m.n) the poset-metric d(p) introduced by Rosenbloom and Tsfasman. In this paper, we give a complete description of the symmetry group of the metric space (V, d(p)). (C) 2008 Elsevier B.V. All rights reserved.Let F(q)(m.n) he the vector space of m . n-tuples over a finite field Fit and P = {1, 2,..., m - n} a poset that is the finite union of In disjoint chains of length it. We consider on F(q)(m.n) the poset-metric d(p) introduced by Rosenbloom and Tsfasman.3094763771FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOsem informaçãoThe authors would like to thank the anonymous referee for valuable remarks which led to sensible improvements in the text and in the proof

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

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

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

Near MDS poset codes and distributions

We study $q$-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