Advances in Mathematics of Communications (AMC)

Canonical- systematic form for codes in hierarchical poset metrics

Pages: 315 - 328, Volume 6, Issue 3, August 2012      doi:10.3934/amc.2012.6.315

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Luciano Viana Felix - UFRRJ - Universidade Federal Rural do Rio de Janeiro, BR467, km7, 23890-000 Serop├ędica - RJ, Brazil (email)
Marcelo Firer - IMECC/UNICAMP - State University of Campinas, Rua Srgio Buarque de Holanda, 651, Cidade Universitria 'Zeferino Vaz, 13083-859 - Campinas - SP, Brazil (email)

Abstract: In this work we present a canonical-systematic form of a generator matrix for linear codes whith respect to a hierarchical poset metric on the linear space $\mathbb F_q^n$. We show that up to a linear isometry any such code is equivalent to the direct sum of codes with smaller dimensions. The canonical-systematic form enables to exhibit simple expressions for the generalized minimal weights (in the sense defined by Wei), the packing radius of the code, characterization of perfect codes and also syndrome decoding algorithm that has (in general) exponential gain when compared to usual syndrome decoding.

Keywords:  Poset codes, generalized Hamming weights, hierarchical posts.
Mathematics Subject Classification:  Primary: 94B05, 94B75; Secondary: 05B25.

Received: September 2011;      Revised: May 2012;      Available Online: August 2012.