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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


IMPACT FACTOR 2018: 0.789

CiteScore 2018: 0.73

SCImago Journal Rank (SJR) 2018: 0.329
Source Normalized Impact per Paper (SNIP) 2018: 0.908

Mathematical Citation Quotient (MCQ) 2018: 0.53

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On plane curves given by separated polynomials and their automorphisms

Matteo Bonini / Maria Montanucci / Giovanni Zini
Published Online: 2019-06-20 | DOI: https://doi.org/10.1515/advgeom-2019-0005

Abstract

Let 𝓒 be a plane curve defined over the algebraic closure K of a finite prime field 𝔽p by a separated polynomial, that is 𝓒 : A(Y) = B(X), where A(Y) is an additive polynomial of degree pn and the degree m of B(X) is coprime with p. Plane curves given by separated polynomials are widely studied; however, their automorphism groups are not completely determined. In this paper we compute the full automorphism group of 𝓒 when m ≢ 1 mod pn and B(X) = Xm. Moreover, some sufficient conditions for the automorphism group of 𝓒 to imply that B(X) = Xm are provided. Also, the full automorphism group of the norm-trace curve 𝓒 : X(qr – 1)/(q–1) = Yqr–1 + Yqr–2 + … + Y is computed. Finally, these results are used to show that certain one-point AG codes have many automorphisms.

Keywords: Plane curve; separated polynomial; AG code; code automorphisms

MSC 2010: 14H05; 14H37; 94B27

References

  • [1]

    E. Ballico, A. Ravagnani, On the duals of geometric Goppa codes from norm-trace curves. Finite Fields Appl. 20 (2013), 30–39. MR3015349 Zbl 1308.94112Web of ScienceCrossrefGoogle Scholar

  • [2]

    D. Bartoli, M. Montanucci, G. Zini, Multi point AG codes on the GK maximal curve. Des. Codes Cryptogr. 86 (2018), 161–177. MR3742839 Zbl 06830015CrossrefGoogle Scholar

  • [3]

    O. Geil, On codes from norm-trace curves. Finite Fields Appl. 9 (2003), 351–371. MR1983054 Zbl 1032.94015CrossrefWeb of ScienceGoogle Scholar

  • [4]

    M. Giulietti, G. Korchmáros, On automorphism groups of certain Goppa codes. Des. Codes Cryptogr. 47 (2008), 177–190. MR2375466 Zbl 1178.94245CrossrefGoogle Scholar

  • [5]

    H.-W. Henn, Funktionenkörper mit grosser Automorphismengruppe. J. Reine Angew. Math. 302 (1978), 96–115. MR511696 Zbl 0378.12011Google Scholar

  • [6]

    J. W. P. Hirschfeld, G. Korchmáros, F. Torres, Algebraic curves over a finite field. Princeton Univ. Press 2008. MR2386879 Zbl 1200.11042Google Scholar

  • [7]

    B. Huppert, Endliche Gruppen. I. Springer 1967. MR0224703 Zbl 0217.07201Google Scholar

  • [8]

    D. Joyner, A. Ksir, Automorphism groups of some AG codes. IEEE Trans. Inform. Theory 52 (2006), 3325–3329. MR2240022 Zbl 1225.94029CrossrefGoogle Scholar

  • [9]

    S. Miura, N. Kamiya, Geometric Goppa codes on some maximal curves and their minimum distance. Proc. IEEE Workshop on Information Theory, Susono-shi, Japan (1993), 85–86.Google Scholar

  • [10]

    C. Munuera, A. Sepulveda, F. Torres, Algebraic Geometry codes from Castle curves. Coding Theory and Applications. Springer, Berlin, Heidelberg, (2008), 117–127. Zbl 1166.94012Google Scholar

  • [11]

    C. Munuera, G. Tizziotti, F. Torres, Two-point codes on Norm-Trace curves. Coding Theory and Applications, (2008), 128–136. Zbl 1166.94356Google Scholar

  • [12]

    H. Stichtenoth, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. II. Ein spezieller Typ von Funktionenkörpern. Arch. Math. (Basel) 24 (1973), 615–631. MR0404265 Zbl 0282.14007CrossrefGoogle Scholar

  • [13]

    H. Stichtenoth, Algebraic function fields and codes. Springer 2009. MR2464941 Zbl 1155.14022Google Scholar

  • [14]

    M. A. Tsfasman, S. G. Vlăduţ, Algebraic-geometric codes, volume 58 of Mathematics and its Applications (Soviet Series). Kluwer 1991. MR1186841 Zbl 0727.94007Google Scholar

About the article

Received: 2017-08-21

Revised: 2018-02-13

Published Online: 2019-06-20


Communicated by: G. Korchmáros


Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2019-0005.

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