Poset metrics admitting association schemes and a new proof of MacWilliams identity
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It is known that being hierarchical is a necessary and sufficient condition for a poset to admit MacWilliams identity. In this paper, we completely characterize the structures of posets which have an association scheme structure whose relations are indexed by the poset distance between the points in the space. We also derive an explicit formula for the eigenmatrices of association schemes induced by such posets. By using the result of Delsarte which generalizes the MacWilliams identity for linear codes, we give a new proof of the MacWilliams identity for hierarchical linear poset codes.
Abstract
It is known that being hierarchical is a necessary and sufficient condition for a poset to admit MacWilliams identity. In this paper, we completely characterize the structures of posets which have an association scheme structure whose relations are indexed by the poset distance between the points in the space. We also derive an explicit formula for the eigenmatrices of association schemes induced by such posets. By using the result of Delsarte which generalizes the MacWilliams identity for linear codes, we give a new proof of the MacWilliams identity for hierarchical linear poset codes.
참고문헌 (23)
- E. Bannai, "Algebraic Combinatorics. I, Association schemes", pp.1984, The Benjamin
- R. C. Bose, "On linear associative algebras corresponding to associa- tion schemes of partially balanced designs", Ann. Math. Statist, 30, pp.21~38 (1959)
- R. C. Bose, "Classification and analysis of partially balanced in- complete block designs with two associate classes", J. Amer. Statist. Assoc, 47, pp.151~184 (1952)
- A. E. Brouwer, "Distance-Regular Graphs" (1989), Springer-Verlag
- R. A. Brualdi, "Codes with a poset metric", Discrete Math, 147(1-3), pp.57~72 (1995)
- P. Delsarte, "An algebraic approach to the association schemes of coding theory", Philips Res. Repts. Suppl, 10, pp.134~ (1976)
- P. Delsarte, "Association schemes and coding theory", IEEE Trans. Inform. Theory, 44(6), pp.2477~2504 (1998)
- S. T. Dougherty, "MacWilliams duality and the Rosenbloom- Tsfasman metric", Moscow Math. J, 2(1), pp.81~97 (2002)
- J. N. Guti´errez, "A MacWilliams identity for poset-codes", Congr. Number, 133, pp.63~73 (1998)
- Y. Jang, "On a MacWilliams identity and a perfectness for a binary linear (n, n − 1, j)-poset code", Discrete Math, 265(1-3), pp.85~104 (2003)
- D. S. Kim, "A MacWilliams-type identity for linear codes on weak order", Discrete Math, 263(1-3), pp.181~194 (2003)
- H. K. Kim, "A classification of posets admitting the MacWilliams identity", IEEE Trans. Inform. Theory, 51(4), pp.1424~1431 (2005)
- R. Lidl, "Introduction to Finite Fields and Their Applications" (1994), Cambridge University Press
- F. J. MacWilliams, "The Theory of Error-Correcting Codes" (1977), North- Holland
- W. J. Martin, "Association schemes for ordered orthogonal arrays and (T,M, S)-nets", Canad. J. Math, 51(2), pp.326~346 (1999)
- H. Niederreiter, "Point sets and sequence with small discrepancy", Monatsh. Math, 104(4), pp.273~337 (1987)
- H. Niederreiter, "A combinatorial problem for vector spaces over finite fields", Discrete Math, 96(3), pp.221~228 (1991)
- H. Niederreiter, "Orthogonal arrays and other combinatorial aspects in the theory of uniform point distributions in unit cubes", Discrete Math, 106, pp.107 (1992), 361~367
- M. Y. Rosenbloom, "Codes for m-metric", Probl. Pered. Inf, 33, pp.55~63 (1997)
- M. Y. Rosenbloom, "Codes for m-metric", English translation: Probl. Inform. Transm, 33, pp.45~52 (1997)
- M. M. Skigrnov, "Coding theory and uniform distibutions", Algebra i Analiz, 13, pp.191~239 (2001)
- M. M. Skigrnov, "Coding theory and uniform distibutions", English translation: St. Petesburg Math. J, 13, pp.301~337 (2002)
- O. Tamaschke, "Zur Theorie der Permutationsgruppen mit regul¨arer Untergruppe. I, II", Math. Z, 80, pp.328~354 (1963)