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

Nazar Arakelian; Saeed Tafazolian; Fernando Torres

doi:10.3934/amc.2018009

Article Contents

2018, Volume 12, Issue 1: 143-149. Doi: 10.3934/amc.2018009

This issue Previous Article Trace description and Hamming weights of irreducible constacyclic codes Next Article Channel decomposition for multilevel codes over multilevel and partial erasure channels

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

1.
Centro de Matemática, Computação e Cognição (CMCC), Universidade Federal do ABC, Avenida dos Estados 5001, 09210-580, Santo André, SP , Brazil
2.
School of Mathematics, Institute for Research in Fundamental Science (IPM), P.O. Box 19395-5746, Tehran, Iran
3.
Dept. of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave, P.O. Box: 15875-4413, Tehran, Iran
4.
Instituto de Matemática, Estatística e Computação Científica (IMECC), Universidade Estadual de Campinas, R. Sérgio Buarque de Holanda 651, Cidade Universitária "Zeferino Vaz", 13083-859, Campinas, SP, Brazil

Received: September 30, 2016

Published: February 2018

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	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.