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Designs, Codes and Cryptography

Erschienen in:

27.04.2020

Weierstrass semigroup at \(m+1\) rational points in maximal curves which cannot be covered by the Hermitian curve

verfasst von: Alonso Sepúlveda Castellanos, Maria Bras-Amorós

Erschienen in: Designs, Codes and Cryptography | Ausgabe 8/2020

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Abstract

We determine the Weierstrass semigroup \(H(P_\infty ,P_1,\ldots ,P_m)\) at several rational points on the maximal curves which cannot be covered by the Hermitian curve introduced in Tafazolian et al. (J Pure Appl Algebra 220(3):1122–1132, 2016). Furthermore, we present some conditions to find pure gaps. We use this semigroup to obtain AG codes with better relative parameters than comparable one-point AG codes arising from these curves.

Vorheriger Artikel Power error locating pairs

Nächster Artikel Isometry-dual flags of AG codes

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Bartoli D., Masuda A.M., Montanucci M., Quoos L.: Pure gaps on curves with many rational places. Finite Fields Appl. 53, 287–308 (2018). https://doi.org/10.1016/j.ffa.2018.07.001.MathSciNetCrossRefMATH

Bartoli D., Montanucci M., Zini G.: AG codes and AG quantum codes from GGS curves. Des. Codes Cryptogr. 86(10), 2315–2344 (2018).MathSciNetCrossRef

Bartoli D., Montanucci M., Zini G.: Multi-point AG codes on the GK maximal curve. Des. Codes Cryptogr. 86(1), 161–177 (2018).MathSciNetCrossRef

Beelen P., Montanucci M.: A new family of maximal curves. J. Lond. Math. Soc. 98, 573–592 (2018).MathSciNetCrossRef

Bras-Amorós M.: On numerical semigroups and the redundancy of improved codes correcting generic errors. Des. Codes Cryptogr. 53(2), 111–118 (2009). https://doi.org/10.1007/s10623-009-9297-8.MathSciNetCrossRefMATH

Bras-Amorós M., O’Sullivan M.E.: The correction capability of the Berlekamp-Massey-Sakata algorithm with majority voting. Appl. Algebra Energy Commun. Comput. 17(5), 315–335 (2006). https://doi.org/10.1007/s00200-006-0015-8.MathSciNetCrossRefMATH

Bras-Amorós M., Lee K., Vico-Oton A.: New lower bounds on the generalized Hamming weights of AG codes. IEEE Trans. Inform. Theory 60(10), 5930–5937 (2014). https://doi.org/10.1109/TIT.2014.2343993.MathSciNetCrossRefMATH

Campillo A., Farrán J.I.: Computing Weierstrass semigroups and the Feng-Rao distance from singular plane models. Finite Fields Appl. 6(1), 71–92 (2000). https://doi.org/10.1006/ffta.1999.0266.MathSciNetCrossRefMATH

Campillo A., Farrán J.I., Munuera C.: On the parameters of algebraic-geometry codes related to Arf semigroups. IEEE Trans. Inform. Theory 46(7), 2634–2638 (2000). https://doi.org/10.1109/18.887872.MathSciNetCrossRefMATH

10.

Carvalho C., Torres F.: On Goppa codes and Weierstrass gaps at several points. Des. Codes Cryptogr. 35(2), 211–225 (2005).MathSciNetCrossRef

11.

Castellanos A.S., Tizziotti G.: Two-points AG codes on the GK maximal curves. IEEE Trans. Inform. Theory 62(9), 4867–4872 (2016). https://doi.org/10.1109/TIT.2015.2511787.MathSciNetCrossRefMATH

12.

Castellanos A.S., Tizziotti G.: Weierstrass semigroup and pure gaps at several points on the GK curve. Bull. Braz. Math. Soc. (N.S.) 49(2), 419–429 (2018). https://doi.org/10.1007/s00574-017-0059-3.MathSciNetCrossRefMATH

13.

Duursma I.M., Park S.: Delta sets for divisors supported in two points. Finite Fields Appl. 18(5), 865–885 (2012). https://doi.org/10.1016/j.ffa.2012.06.005.MathSciNetCrossRefMATH

14.

Fanali S., Giulietti M.: One-point AG codes on the GK maximal curves. IEEE Trans. Inform. Theory 56(1), 202–210 (2010).MathSciNetCrossRef

15.

Farrán J.I., Munuera C.: Goppa-like bounds for the generalized Feng-Rao distances. Discret. Appl. Math. 128(1), 145–156 (2003). https://doi.org/10.1016/S0166-218X(02)00441-9. International Workshop on Coding and Cryptography (WCC 2001) (Paris)

16.

Feng G.L., Rao T.R.N.: A simple approach for construction of algebraic-geometric codes from affine plane curves. IEEE Trans. Inform. Theory 40(4), 1003–1012 (1994). https://doi.org/10.1109/18.335972.MathSciNetCrossRefMATH

17.

Fuhrmann R., Garcia A., Torres F.: On maximal curves. J. Number Theory 67(1), 29–51 (1997).MathSciNetCrossRef

18.

Fulton W.: Algebraic curves. An introduction to algebraic geometry. W. A. Benjamin Inc., New York-Amsterdam (1969). Notes written with the collaboration of Richard Weiss, Mathematics Lecture Notes Series

19.

Garcia A., Stichtenoth H.: A maximal curve which is not a galois subcover of the hermitian curve. Bull. Braz. Math. Soc. (N.S.) 37(1), 139–152 (2006).MathSciNetCrossRef

20.

Garcia A., Güneri C., Stichtenoth H.: A generalization of the giulietti-korchmáros maximal curve. Adv. Geom. 10(3), 427–434 (2010).MathSciNetCrossRef

21.

Geil O., Matsumoto R.: Bounding the number of \(\mathbb{F}_q\)-rational places in algebraic function fields using Weierstrass semigroups. J. Pure Appl. Algebra 213(6), 1152–1156 (2009). https://doi.org/10.1016/j.jpaa.2008.11.013.MathSciNetCrossRefMATH

22.

Giulietti M., Korchmáros G.: A new family of maximal curves over a finite field. Math. Ann. 343, 229–245 (2009).MathSciNetCrossRef

23.

Goppa V.D.: Codes on algebraic curves. Dokl. Akad. NAUK SSSR 259, 1289–1290 (1981).MathSciNetMATH

24.

Goppa V.D.: Algebraic-geometric codes. Izv. Akad. NAUK SSSR 46, 75–91 (1982).MathSciNetMATH

25.

Heijnen P., Pellikaan R.: Generalized Hamming weights of \(q\)-ary Reed-Muller codes. IEEE Trans. Inform. Theory 44(1), 181–196 (1998). https://doi.org/10.1109/18.651015.MathSciNetCrossRefMATH

26.

Høholdt T., van Lint J.H., Pellikaan R.: Algebraic geometry codes. In: Handbook of coding theory, Vol. I, II, pp. 871–961. North-Holland, Amsterdam (1998).

27.

Homma M., Kim S.J.: Goppa codes with Weierstrass pairs. J. Pure Appl. Algebra 162(2–3), 273–290 (2001).MathSciNetCrossRef

28.

Hu C., Yang S.: Multi-point codes from GGS curves. Adv. Math. Commun. 14, 279 (2020). https://doi.org/10.3934/amc2020020.CrossRefMATH

29.

Kirfel C., Pellikaan R.: The minimum distance of codes in an array coming from telescopic semigroups. IEEE Trans. Inform. Theory 41(6 part I), 1720–1732 (1995). https://doi.org/10.1109/18.476245.MathSciNetCrossRefMATH

30.

Korchmáros G., Torres F.: On the genus of a maximal curve. Math. Ann. 323(3), 589–608 (2002).MathSciNetCrossRef

31.

Lachaud G.: Sommes d’Eisenstein et nombre de points de certaines courbes algébriques sur les corps finis. CR. Acad. Sci 305, 729–732 (1987).MATH

32.

Matthews G.L.: The Weierstrass semigroup of an \(m\)-tuple of collinear points on a Hermitian curve. In: Finite fields and applications, Lecture Notes in Comput. Sci., Vol. 2948, pp. 12–24. Springer, Berlin (2004)

33.

Pellikaan R., Torres F.: On Weierstrass semigroups and the redundancy of improved geometric Goppa codes. IEEE Trans. Inform. Theory 45(7), 2512–2519 (1999). https://doi.org/10.1109/18.796393.MathSciNetCrossRefMATH

34.

Pellikaan R., Stichtenoth H., Torres F.: Weierstrass semigroups in an asymptotically good tower of function fields. Finite Fields Appl. 4(4), 381–392 (1998).MathSciNetCrossRef

35.

Schmid W.C., Schürer R.: MinT. http://mint.sbg.ac.at/

36.

Tafazolian S., Teherán-Herrera A., Torres F.: Further examples of maximal curves which cannot be covered by the Hermitian curve. J. Pure Appl. Algebra 220(3), 1122–1132 (2016). https://doi.org/10.1016/j.jpaa.2015.08.010.MathSciNetCrossRefMATH

Titel: Weierstrass semigroup at rational points in maximal curves which cannot be covered by the Hermitian curve
verfasst von: Alonso Sepúlveda Castellanos
Maria Bras-Amorós
Publikationsdatum: 27.04.2020
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 8/2020
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-020-00757-4

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