\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On the spectrum for the genera of maximal curves over small fields

Abstract Full Text(HTML) Related Papers Cited by
  • Motivated by previous computations in Garcia, Stichtenoth and Xing (2000) paper [11], we discuss the spectrum $\mathbf{M}(q^2)$ for the genera of maximal curves over finite fields of order $q^2$ with $7≤ q≤ 16$. In particular, by using a result in Kudo and Harashita (2016) paper [22], the set $\mathbf{M}(7^2)$ is completely determined.

    Mathematics Subject Classification: Primary: 11G20, 11M38, 14H05, 14G15; Secondary: 14HXX.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  •   manYPoints-Table of Curves with Many Points, available at www.manypoints.org
      M. Abdón  and  F. Torres , Maximal curves in characteristic two, Manuscripta Math., 99 (1999) , 39-53. 
      A. Cossidente , G. Korchmáros  and  F. Torres , On curves covered by the Hermitian curve, J. Algebra, 216 (1999) , 56-76. 
      Y. Danisman  and  M. Ozdemir , On the genus spectrum of maximal curves over finite fields, J. Discrete Math. Sci. Crypt., 18 (2015) , 513-529. 
      I. Duursma  and  K. H. Mak , On maximal curves which are not Galois subcovers of the Hermitian curve, Bull. Braz. Math. Soc. New Series, 43 (2012) , 453-465. 
      S. Fanali  and  M. Giulietti , On some open problems on maximal curves, Des. Codes Crypt., 56 (2010) , 131-139. 
      S. Fanali , M. Giulietti  and  I. Platoni , On maximal curves over finite fields of small order, Adv. Math. Commun., 6 (2012) , 107-120. 
      R. Fuhrmann , A. Garcia  and  F. Torres , On maximal curves, J. Number Theory, 67 (1997) , 29-51. 
      R. Fuhrmann  and  F. Torres , The genus of curves over finite fields with many rational points, Manuscripta Math., 89 (1996) , 103-106. 
      A. Garcia  and  H. Stichtenoth , A maximal curve which is not a Galois subcover of the Hermitian curve, Bulletin Braz. Math. Soc., 37 (2006) , 1-14. 
      A. Garcia , H. Stichtenoth  and  C. P. Xing , On subfields of the Hermitian function field, Compositio Math., 120 (2000) , 137-170. 
      M. Giulietti  and  G. Korchmáros , A new family of maximal curves over a finite field, Math. Ann., 343 (2009) , 229-245. 
      M. Giulietti, M. Montanucci, L. Quoos and G. Zini, The automorphism group of some Galois covers of the Suzuki and Ree curves, preprint, arXiv: 1609.09343
      M. Giulietti , M. Montanucci  and  G. Zini , On maximal curves that are not quotients of the Hermitian curve, Finite Fields Appl., 41 (2016) , 71-78. 
      M. Giulietti , L. Quoos  and  G. Zini , Maximal curves from subcovers of the GK-curve, J. Pure Appl. Algebra, 220 (2016) , 3372-3383. 
      J. W. P. Hirschfeld, G. Korchmáros and F. Torres, Algebraic Curves over Finite Fields, Princeton Univ. Press, 2008.
      E. W. Howe, Quickly constructing curves of genus 4 with many points, in Frobenius Distributions: Sato-Tate and Lang-Trotter Conjectures, Amer. Math. Soc., Providence, 2016, 149–173.
      N. E. Hurt, Many Rational Points, Coding Theory and Algebraic Geometry, Kluwer Acad. Publ., The Netherlands, 2003.
      I. Ihara , Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Tokyo Sec. Ia, 28 (1981) , 721-724. 
      G. Korchmáros  and  F. Torres , Embedding of a maximal curve in a Hermitian variety, Compositio Math., 128 (2001) , 95-113. 
      G. Korchmáros  and  F. Torres , On the genus of a maximal curve, Math. Ann., 323 (2002) , 589-608. 
      M. Kudo  and  S. Harashita , Superspecial curves of genus 4 in small characteristic, Finite Fields Appl., 45 (2017) , 131-169. 
      M. Montanucci and G. Zini, On the spectrum of genera of quotients of the Hermitian curve, preprint, arXiv: 1703.10592
      M. Montanucci  and  G. Zini , Some Ree and Suzuki curves are not Galois covered by the Hermitian curve, Finite Fields Appl., 48 (2017) , 175-195. 
      H. G. Rück  and  H. Stichtenoth , A characterization of Hermitian function fields over finite fields, J. Reine Angew. Math., 457 (1994) , 185-188. 
      H. Stichtenoth, Algebraic Function Fields and Codes, 2nd edition, Springer-Verlag, New York, 2009.
      K. O. Stöhr  and  J. F. Voloch , Weierstrass points and curves over finite fields, Proc. London Math. Soc., 52 (1986) , 1-19. 
      S. Tafazolian , A. Teherán-Herrera  and  F. Torres , Further examples of maximal curves which cannot be covered by the Hermitian curve, J. Pure Appl. Algebra, 220 (2016) , 1122-1132. 
      S. Tafazolian  and  F. Torres , On the curve $y^n=x^m+x$ over finite fields, J. Number Theory, 145 (2014) , 51-66. 
      C. P. Xing  and  H. Stichtenoth , The genus of maximal functions fields, Manuscripta Math., 86 (1995) , 217-224. 
  • 加载中
SHARE

Article Metrics

HTML views(740) PDF downloads(389) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return