On the Minimum Distance of One-Point Castle AG Codes
DOI:
https://doi.org/10.17230/ingciencia.8.16.9Keywords:
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
Downloads
References
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
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
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
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
V. Goppa, “Codes on algebraic curves”, Sov. Math.-Dokl., vol. 24, pp. 170-172,1981. Referenciado en 239
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
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
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
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
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
J. van Lint, Introduction to Coding Theory, 2.a ed. Springer-Verlag, 1992. Referenciadoen 239
J. van Lint y G. V. D. Geer, Introduction to coding theory and algebraic geometry.Birkhauser Verlag, 1988. Referenciado en 239
“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
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
C. Shannon, “A mathematical theory of communication”, Bell System TechnicalJournal, vol. 27, pp. 656-715, 1948. Referenciado en 238
H. Stichtenoth, Algebraic Function Fields and Codes. Springer-Verlag, 2009.Referenciado en 240, 245
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
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
Downloads
Published
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
Plagiarism Detection