On the Minimum Distance of One-Point Castle AG Codes
Main Article Content
Keywords
Error-correcting codes, AG codes, minimum distance, Hermitian codes.
Abstract
We present a characterization of the lower bound d* for minimum distance of algebraic geometry one-point codes coming from castle curves. This article shows explicit calculations of this bound in the case of a Weierstrass semigroup generated by two consecutive elements. In particular, we obtain a simple characterization of the exact value of the minimum distance Hermitian codes.
MSC: 94.B27; 94.B65
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References
[1] A. Garcia, H. Stichtenoth, “A Class of Polynomials over Finite Fields”, FiniteFields and Their Applications, vol. 5, n.o 4, pp. 424-435, oct. 1999. Referenciado en 241
[2] O. Geil, “On codes from norm-trace curves”, Finite Fields and Their Applications,vol. 9, n.o 3, pp. 351-371, jul. 2003. Referenciado en 241
[3] F. Torres, D. Ruano, C. Munuera, O. Geil, “On the order bounds for one-point AGcodes”, Advances in Mathematics of Communications, vol. 5, n.o 3, pp. 489-504,ago. 2011. Referenciado en 241, 243
[4] O. Geil, R. Matsumoto, “Bounding the number of -rational places in algebraicfunction fields using Weierstrass semigroups”, Journal of Pure and Applied Algebra,vol. 213, n.o 6, pp. 1152-1156, jun. 2009. Referenciado en 241, 242
[5] V. Goppa, “Codes on algebraic curves”, Sov. Math.-Dokl., vol. 24, pp. 170-172,1981. Referenciado en 239
[6] J. Hansen, “Deligne-Lusztig varieties and group codes”, in Coding Theory and AlgebraicGeometry, vol. 1518, H. Stichtenoth y M. Tsfasman, Eds. Springer Berlin/ Heidelberg, 1992, pp. 63-81. Referenciado en 241
[7] J. Pedersen, J. Hansen, “Automorphism groups of Ree type Deligne-Lusztig curvesand function fields.”, J. reine angew. Math. , n.o 440, pp. 99-109, 1993.Referenciado en 241
[8] J. Hansen, H. Stichtenoth, “Group codes on certain algebraic curves with manyrational points”, Applicable Algebra in Engineering, Communication and Computing,vol. 1, n.o 1, pp. 67-77, 1990. Referenciado en 241
[9] T. Høholdt, J. van Lint, R. Pellikaan, “Algebraic-geometry codes”, in Handbookof Coding Theory, Volume 1: Part 1: Algebraic Coding, V. Pless y W. . Huffman,Eds. Amsterdam: Elsevier, 1998, pp. 871-961. Referenciado en 242, 251
[10] J. Lewittes, “Places of degree one in function fields over finite fields”, Journal ofPure and Applied Algebra, vol. 69, n.o 2, pp. 177-183, dic. 1990.Referenciado en 241, 242
[11] J. van Lint, Introduction to Coding Theory, 2.a ed. Springer-Verlag, 1992. Referenciadoen 239
[12] J. van Lint y G. V. D. Geer, Introduction to coding theory and algebraic geometry.Birkhauser Verlag, 1988. Referenciado en 239
[13] “MinT”, Online database for optimal parameters of (t, m, s)-nets,(t, s)-secuences, ortogonal arrays and linear codes.
[Online]. Available:http://mint.sbg.ac.at/. Referenciado en 251
[14] C. Munuera, A. Sepúlveda, F. Torres, “Algebraic Geometry Codes from CastleCurves”, in Coding Theory and Applications, vol. 5228, Á. Barbero, Ed. Berlin,Heidelberg: Springer Berlin Heidelberg, 2008, pp. 117-127.Referenciado en 241, 242
[15] C. Shannon, “A mathematical theory of communication”, Bell System TechnicalJournal, vol. 27, pp. 656-715, 1948. Referenciado en 238
[16] H. Stichtenoth, Algebraic Function Fields and Codes. Springer-Verlag, 2009.Referenciado en 240, 245
[17] M. Tsfasman, S. Vladutx, T. Zink, “Modular curves, Shimura curves, and Goppacodes, better than Varshamov-Gilbert bound”, Mathematische Nachrichten, vol.109, n.o 1, pp. 21-28, 1982. Referenciado en 240
[18] K. Yang, P. Kumar, “On the true minimum distance of Hermitian codes”, in CodingTheory and Algebraic Geometry, vol. 1518, H. Stichtenoth y M. A. Tsfasman,Eds. Springer Berlin Heidelberg, pp. 99-107. Referenciado en 250, 251
[2] O. Geil, “On codes from norm-trace curves”, Finite Fields and Their Applications,vol. 9, n.o 3, pp. 351-371, jul. 2003. Referenciado en 241
[3] F. Torres, D. Ruano, C. Munuera, O. Geil, “On the order bounds for one-point AGcodes”, Advances in Mathematics of Communications, vol. 5, n.o 3, pp. 489-504,ago. 2011. Referenciado en 241, 243
[4] O. Geil, R. Matsumoto, “Bounding the number of -rational places in algebraicfunction fields using Weierstrass semigroups”, Journal of Pure and Applied Algebra,vol. 213, n.o 6, pp. 1152-1156, jun. 2009. Referenciado en 241, 242
[5] V. Goppa, “Codes on algebraic curves”, Sov. Math.-Dokl., vol. 24, pp. 170-172,1981. Referenciado en 239
[6] J. Hansen, “Deligne-Lusztig varieties and group codes”, in Coding Theory and AlgebraicGeometry, vol. 1518, H. Stichtenoth y M. Tsfasman, Eds. Springer Berlin/ Heidelberg, 1992, pp. 63-81. Referenciado en 241
[7] J. Pedersen, J. Hansen, “Automorphism groups of Ree type Deligne-Lusztig curvesand function fields.”, J. reine angew. Math. , n.o 440, pp. 99-109, 1993.Referenciado en 241
[8] J. Hansen, H. Stichtenoth, “Group codes on certain algebraic curves with manyrational points”, Applicable Algebra in Engineering, Communication and Computing,vol. 1, n.o 1, pp. 67-77, 1990. Referenciado en 241
[9] T. Høholdt, J. van Lint, R. Pellikaan, “Algebraic-geometry codes”, in Handbookof Coding Theory, Volume 1: Part 1: Algebraic Coding, V. Pless y W. . Huffman,Eds. Amsterdam: Elsevier, 1998, pp. 871-961. Referenciado en 242, 251
[10] J. Lewittes, “Places of degree one in function fields over finite fields”, Journal ofPure and Applied Algebra, vol. 69, n.o 2, pp. 177-183, dic. 1990.Referenciado en 241, 242
[11] J. van Lint, Introduction to Coding Theory, 2.a ed. Springer-Verlag, 1992. Referenciadoen 239
[12] J. van Lint y G. V. D. Geer, Introduction to coding theory and algebraic geometry.Birkhauser Verlag, 1988. Referenciado en 239
[13] “MinT”, Online database for optimal parameters of (t, m, s)-nets,(t, s)-secuences, ortogonal arrays and linear codes.
[Online]. Available:http://mint.sbg.ac.at/. Referenciado en 251
[14] C. Munuera, A. Sepúlveda, F. Torres, “Algebraic Geometry Codes from CastleCurves”, in Coding Theory and Applications, vol. 5228, Á. Barbero, Ed. Berlin,Heidelberg: Springer Berlin Heidelberg, 2008, pp. 117-127.Referenciado en 241, 242
[15] C. Shannon, “A mathematical theory of communication”, Bell System TechnicalJournal, vol. 27, pp. 656-715, 1948. Referenciado en 238
[16] H. Stichtenoth, Algebraic Function Fields and Codes. Springer-Verlag, 2009.Referenciado en 240, 245
[17] M. Tsfasman, S. Vladutx, T. Zink, “Modular curves, Shimura curves, and Goppacodes, better than Varshamov-Gilbert bound”, Mathematische Nachrichten, vol.109, n.o 1, pp. 21-28, 1982. Referenciado en 240
[18] K. Yang, P. Kumar, “On the true minimum distance of Hermitian codes”, in CodingTheory and Algebraic Geometry, vol. 1518, H. Stichtenoth y M. A. Tsfasman,Eds. Springer Berlin Heidelberg, pp. 99-107. Referenciado en 250, 251