Canonical- Systematic Form For Codes In Hierarchical Poset Metrics

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 F n q. 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. © 2012 AIMS.63315328Alves, M.M.S., Panek, L., Firer, M., Error-block codes and poset metrics (2008) Adv. Math. Commun., 2, pp. 95-111Alves, M.M.S., Panek, L., Firer, M., Classification of Niederreiter-Rosenbloom-Tsfasmanblock codes (2010) IEEE Trans. Inform. Theory, 56, pp. 5207-5216Brualdi, R., Graves, J.S., Lawrence, M., Codes with a poset metric (1995) Discrete Math., 147, pp. 57-72Hardy, G.H., Ramanujan's work on partitions and Asymptotic theory of partitions (1999) Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, pp. 83-100+113-131. , Chs. 6and 8 in 3 rd edition, Chelsea, New YorkKim, H.K., Oh, D.Y., A classification of posets admitting the MacWilliams identity (2005) IEEE Trans. Inform. Theory, 51, pp. 1424-1431Niederreiter, H., A combinatorial problem for vector spaces over finite fields (1991) Discrete Math., 96, pp. 221-228Niederreiter, H., Orthogonal array and other combinatorial aspects in the theory of uniformpoint distributions in unit cubes (1992) Discrete Math, 106-107, pp. 361-367Panek, L., Firer, M., Kim, H.K., Hyun, J.Y., Groups of linear isometries on poset struc-tures (2008) Discrete Math., 308, pp. 4116-4123Rosenbloom, M.Y., Tsfasman, M.A., Codes for the m-metric (1997) Prob. Inform. Transm., 33, pp. 45-52Wei, V.K., Generalized Hamming weights for linear codes (1991) IEEE Trans. Inform. Theory, 37, pp. 1412-141