Abstract
The quality of an algebraic geometry code depends on the curve from which the code has been defined. In this paper we consider codes obtained from Castle curves, namely those whose number of rational points attains the Lewittes’ bound for some rational point Q and the Weierstrass semigroup at Q is symmetric.
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Abdón, M., Garcia, A.: On a characterization of certain maximal curves. Finite Fields Appl. 10, 133–158 (2004)
Abdón, M., Torres, F.: On maximal curves in characteristic two. Manuscripta Math. 99, 39–53 (1999)
Abdón, M., Torres, F.: On \({\mathbf F}_{q^2}\)-maximal curves of genus q(q − 3)/6. Beitr. Algebra Geom. 46, 241–260 (2005)
Bulygin, S.V.: Generalized Hermitian codes over GF(2r). IEEE Trans. Inform. Theory 52, 4664–4669 (2006)
Carvalho, C., Munuera, C., Silva, E., Torres, F.: Near orders and codes. IEEE Trans. Inform. Theory 53, 1919–1924 (2009)
Deligne, P., Lusztig, G.: Representations of reductive groups over finite fields. Ann. of Math. 103, 103–161 (1976)
Deolalikar, V.: Determining irreducibility and ramification groups for an additive extension of the rational function fields. J. Number. Theory 97, 269–286 (2002)
Fuhrmann, R., Torres, F.: On Weierstrass points and optimal curves. Supplemento ai Rendiconti del Circolo Matematico di Palermo 51, 25–46 (1998)
Garcia, A., Stichtenoth, H.: A class of polynomials over finite fields. Finite Fields Their Applic. 5, 424–435 (1999)
Geil, O.: On codes from norm-trace curves. Finite fields and their Applications 9, 351–371 (2003)
Geil, O., Matsumoto, H.: Bounding the number of rational places using Weierstrass semigroups (preprint, 2007)
Goppa, V.D.: Geometry and Codes. Mathematics and its applications, vol. 24. Kluwer, Dordrecht (1991)
Goppa, V.D.: Codes associated with divisors. Problems Inform. Transmission 13, 22–26 (1977)
Grassl, M.: Bounds on the minimum distance of linear codes, http://www.codetables.de
Hansen, J.P.: Deligne-Lusztig varieties and group codes. Lect. Notes Math. 1518, 63–81 (1992)
Hansen, J.P., Pedersen, J.P.: Automorphism group of Ree type, Deligne-Lusztig curves and function fields. J. Reine Angew. Math. 440, 99–109 (1993)
Hansen, J.P., Stichtenoth, H.: Group codes on certain algebraic curves with many rational points. Applicable Algebra Eng. Comm. Comput. 1, 67–77 (1990)
Henn, H.W.: Funktionenkörper mit grosser Automorphismgruppen. J. Reine Angew Math. 302, 96–115 (1978)
Høholdt, T., van Lint, J.H., Pellikaan, R.: Algebraic-Geometry codes. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, vol. 1. Elsevier, Amsterdam (1998)
Lang, S.: Abelian Varieties. Interscience Pub., New York (1959)
Lewittes, J.: Places of degree one in function fields over finite fields. J. Pure Appl. Algebra 69, 177–183 (1990)
Kirfel, C., Pellikaan, R.: The minimum distance of codes in an array coming from telescopic semigroups. IEEE Trans. Inform. Theory 41, 1720–1732 (1995)
Matthews, G.: Codes from the Suzuki function field. IEEE Trans. Inform. Theory 50, 3298–3302 (2004)
Munuera, C.: On the generalized Hamming weights of geometric Goppa codes. IEEE Trans. Inform. Theory 40(6), 2092–2099 (1994)
Munuera, C., Pellikaan, R.: Equality of geometric Goppa codes and equivalence of divisors. J. Pure Appl. Algebra 90, 229–252 (1993)
Munuera, C., Torres, F.: Bounding the trellis state complexity of algebraic geometric codes. Applicable Algebra Eng. Comm. Computing 15, 81–100 (2004)
Munuera, C., Torres, F.: The structure of algebras admitting well agreeing near weights. J. Pure Appl. Algebra 212, 910–918 (2007)
Pellikaan, R., Shen, B.Z., van Wee, G.J.M.: Which Linear Codes are Algebraic-Geometric. IEEE Trans. Inform. Theory 37, 583–602 (1991)
Sepúlveda, A.: Generalized Hermitian codes over GF(qr) (preprint, 2007)
Stichtenoth, H.: Algebraic Function Fields and Codes. Springer, New York (1993)
Stöhr, K.O., Voloch, J.F.: Weierstrass points and curves over finite fields. Proc. London Math. Soc., 1–19 (1986)
Tsfasman, M.A., Vlǎduţ, S., Zink, T.: Modular curves, Shimura curves and Goppa codes better that Varshamov-Gilbert bound. Math. Nachr. 109, 21–28 (1982)
Tate, J.: Endomorphisms of abelian varieties over finite fields. Inventiones Math. 2, 134–144 (1966)
Yang, K., Kumar, P.V., Stichtenoth, H.: On the weight hierarchy of geometric Goppa codes. IEEE Trans. Inform. Theory 40, 913–920 (1994)
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Munuera, C., Sepúlveda, A., Torres, F. (2008). Algebraic Geometry Codes from Castle Curves. In: Barbero, Á. (eds) Coding Theory and Applications. Lecture Notes in Computer Science, vol 5228. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87448-5_13
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DOI: https://doi.org/10.1007/978-3-540-87448-5_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87447-8
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