# American Institute of Mathematical Sciences

August  2012, 6(3): 315-328. doi: 10.3934/amc.2012.6.315

## Canonical- systematic form for codes in hierarchical poset metrics

 1 UFRRJ - Universidade Federal Rural do Rio de Janeiro, BR467, km7, 23890-000 Seropédica - RJ, Brazil 2 IMECC/UNICAMP - State University of Campinas, Rua Srgio Buarque de Holanda, 651, Cidade Universitria 'Zeferino Vaz, 13083-859 - Campinas - SP, Brazil

Received  September 2011 Revised  May 2012 Published  August 2012

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.
Citation: Luciano Viana Felix, Marcelo Firer. Canonical- systematic form for codes in hierarchical poset metrics. Advances in Mathematics of Communications, 2012, 6 (3) : 315-328. doi: 10.3934/amc.2012.6.315
##### References:
 [1] M. M. S. Alves, L. Panek and M. Firer, Error-block codes and poset metrics, Adv. Math. Commun., 2 (2008), 95-111. doi: 10.3934/amc.2008.2.95.  Google Scholar [2] M. M. S. Alves, L. Panek and M. Firer, Classification of Niederreiter-Rosenbloom-Tsfasman block codes, IEEE Trans. Inform. Theory, 56 (2010), 5207-5216.  Google Scholar [3] R. Brualdi, J. S. Graves and M. Lawrence, Codes with a poset metric, Discrete Math., 147 (1995), 57-72. doi: 10.1016/0012-365X(94)00228-B.  Google Scholar [4] G. H. Hardy, Ramanujan's work on partitions and Asymptotic theory of partitions, Chs. 6 and 8 in "Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work,'' 3rd edition, Chelsea, New York, (1999), 83-100 and 113-131. Google Scholar [5] H. K. Kim and D. Y. Oh, A classification of posets admitting the MacWilliams identity, IEEE Trans. Inform. Theory, 51 (2005), 1424-1431. doi: 10.1109/TIT.2005.844067.  Google Scholar [6] H. Niederreiter, A combinatorial problem for vector spaces over finite fields, Discrete Math., 96 (1991), 221-228. doi: 10.1016/0012-365X(91)90315-S.  Google Scholar [7] H. Niederreiter, Orthogonal array and other combinatorial aspects in the theory of uniform point distributions in unit cubes, Discrete Math., 106/107 (1992), 361-367. doi: 10.1016/0012-365X(92)90566-X.  Google Scholar [8] L. Panek, M. Firer, H. K. Kim and J. Y. Hyun, Groups of linear isometries on poset structures, Discrete Math., 308 (2008), 4116-4123. doi: 10.1016/j.disc.2007.08.001.  Google Scholar [9] M. Y. Rosenbloom and M. A. Tsfasman, Codes for the $m$-metric, Prob. Inform. Transm., 33 (1997), 45-52.  Google Scholar [10] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418. doi: 10.1109/18.133259.  Google Scholar

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##### References:
 [1] M. M. S. Alves, L. Panek and M. Firer, Error-block codes and poset metrics, Adv. Math. Commun., 2 (2008), 95-111. doi: 10.3934/amc.2008.2.95.  Google Scholar [2] M. M. S. Alves, L. Panek and M. Firer, Classification of Niederreiter-Rosenbloom-Tsfasman block codes, IEEE Trans. Inform. Theory, 56 (2010), 5207-5216.  Google Scholar [3] R. Brualdi, J. S. Graves and M. Lawrence, Codes with a poset metric, Discrete Math., 147 (1995), 57-72. doi: 10.1016/0012-365X(94)00228-B.  Google Scholar [4] G. H. Hardy, Ramanujan's work on partitions and Asymptotic theory of partitions, Chs. 6 and 8 in "Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work,'' 3rd edition, Chelsea, New York, (1999), 83-100 and 113-131. Google Scholar [5] H. K. Kim and D. Y. Oh, A classification of posets admitting the MacWilliams identity, IEEE Trans. Inform. Theory, 51 (2005), 1424-1431. doi: 10.1109/TIT.2005.844067.  Google Scholar [6] H. Niederreiter, A combinatorial problem for vector spaces over finite fields, Discrete Math., 96 (1991), 221-228. doi: 10.1016/0012-365X(91)90315-S.  Google Scholar [7] H. Niederreiter, Orthogonal array and other combinatorial aspects in the theory of uniform point distributions in unit cubes, Discrete Math., 106/107 (1992), 361-367. doi: 10.1016/0012-365X(92)90566-X.  Google Scholar [8] L. Panek, M. Firer, H. K. Kim and J. Y. Hyun, Groups of linear isometries on poset structures, Discrete Math., 308 (2008), 4116-4123. doi: 10.1016/j.disc.2007.08.001.  Google Scholar [9] M. Y. Rosenbloom and M. A. Tsfasman, Codes for the $m$-metric, Prob. Inform. Transm., 33 (1997), 45-52.  Google Scholar [10] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418. doi: 10.1109/18.133259.  Google Scholar
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