Linear codes on posets with extension property

FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOWe investigate linear and additive codes in partially ordered Hamming-like spaces that satisfy the extension property, meaning that automorphisms of ideals extend to automorphisms of the poset. The codes are naturally described in terms of translation association schemes that originate from the groups of linear isometries of the space. We address questions of duality and invariants of codes, establishing a connection between the dual association scheme and the scheme defined on the dual poset (they are isomorphic if and only if the poset is self-dual). We further discuss invariants that play the role of weight enumerators of codes in the poset case. In the case of regular rooted trees such invariants are linked to the classical problem of tree isomorphism. We also study the question of whether these invariants are preserved under standard operations on posets such as the ordinal sum and the like. © 2013 Published by Elsevier B.V.We investigate linear and additive codes in partially ordered Hamming-like spaces that satisfy the extension property, meaning that automorphisms of ideals extend to automorphisms of the poset. The codes are naturally described in terms of translation association schemes that originate from the groups of linear isometries of the space. We address questions of duality and invariants of codes, establishing a connection between the dual association scheme and the scheme defined on the dual poset (they are isomorphic if and only if the poset is self-dual). We further discuss invariants that play the role of weight enumerators of codes in the poset case. In the case of regular rooted trees such invariants are linked to the classical problem of tree isomorphism. We also study the question of whether these invariants are preserved under standard operations on posets such as the ordinal sum and the like.3171113FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO2007/56052-8; 2102/20181-7Aho, A., Hopcroft, J., Ullman, J.D., (1974) The Design and Analysis of Computer Algorithms, , Addison-Wesley Publishing Co. Reading, MAAlves, M.M.S., A standard form for generator matrices with respect to the Niederreiter-Rosenbloom-Tsfasman metric (2011) Proc. 2011 IEEE Information Theory Workshop, pp. 486-489. , 16-20 Oct. Paraty, BrazilBarg, A., Firer, M., Translation association schemes and shape enumerators of codes (2012) Proc. 2012 IEEE Internat. Sympos. Information Theory, pp. 101-105. , Boston, MA, JulyBarg, A., Purkayastha, P., Bounds on ordered codes and orthogonal arrays (2009) Mosc. Math. 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