May  2017, 11(2): 289-292. doi: 10.3934/amc.2017020

Certain sextics with many rational points

Department of Mathematics, Shiga University of Medical Science, Seta Tsukinowa-cho, Otsu, Shiga, 520-2192 Japan

Received  February 2016 Revised  March 2016 Published  May 2017

Fund Project: This research was partially supported by JSPS Grant-in-Aid for Young Scientists (B) 25800090.

We construct a family of sextics from the Wiman and Edge sextics. We find a curve over $\mathbb{F}_{5^7}$ attaining the Serre bound, and update $9$ entries of genus $6$ in manYPoints.org by computer search on these sextics.

Citation: Motoko Qiu Kawakita. Certain sextics with many rational points. Advances in Mathematics of Communications, 2017, 11 (2) : 289-292. doi: 10.3934/amc.2017020
References:
[1]

W. Bosma and J. Cannon andC. Playoust, The Magma algebra system. I. The user language, J. Symb. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125. Google Scholar

[2]

W. L. Edge, A pencil of four-nodal plane sextics, Math. Proc. Cambridge Philos. Soc., 89 (1981), 413-421. doi: 10.1017/S0305004100058321. Google Scholar

[3]

A. GarciaG. Güneri and H. Stichtenoth, A generalization of the Giulietti--Korchmáros maximal curve, Adv. Geom., 10 (2010), 427-434. doi: 10.1515/ADVGEOM.2010.020. Google Scholar

[4]

G. van der Geer, E. Howe, K. Lauter and C. Ritzenthaler, Table of curves with many points available at http://www.manypoints.orgGoogle Scholar

[5]

M. GiuliettiM. Montanucci and G. Zini, On maximal curves that are not quotients of the Hermitian curve, Finite Fields Appl, 41 (2016), 72-88. doi: 10.1016/j.ffa.2016.05.005. Google Scholar

[6]

E. Kani and M. Rosen, Idempotent relations and factors of Jacobians, Math. Ann., 284 (1989), 307-327. doi: 10.1007/BF01442878. Google Scholar

[7]

M. Q. Kawakita, Wiman's and Edge's sextic attaining Serre's bound preprint, available at http://www.manypoints.org/upload/kawakita.pdfGoogle Scholar

[8]

M. Q. Kawakita, Wiman's and Edge's sextic attaining Serre's bound Ⅱ, in Algorithmic Arithmetic, Geometry, and Coding Theory, Amer. Math. Soc., 2015, 191–203. doi: 10.1090/conm/637/12758. Google Scholar

[9]

K. Lauter, Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields, J. Algebraic Geom., 10 (2001), 19-36. Google Scholar

[10]

C. Moreno, Algebraic Curves over Finite Fields Cambridge Univ. Press, 1991. doi: 10.1017/CBO9780511608766. Google Scholar

[11]

A. Wiman, Ueber eine einfache Gruppe von 360 ebenen Collineationen, Math. Ann., 47 (1896), 531-556. doi: 10.1007/BF01445800. Google Scholar

show all references

References:
[1]

W. Bosma and J. Cannon andC. Playoust, The Magma algebra system. I. The user language, J. Symb. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125. Google Scholar

[2]

W. L. Edge, A pencil of four-nodal plane sextics, Math. Proc. Cambridge Philos. Soc., 89 (1981), 413-421. doi: 10.1017/S0305004100058321. Google Scholar

[3]

A. GarciaG. Güneri and H. Stichtenoth, A generalization of the Giulietti--Korchmáros maximal curve, Adv. Geom., 10 (2010), 427-434. doi: 10.1515/ADVGEOM.2010.020. Google Scholar

[4]

G. van der Geer, E. Howe, K. Lauter and C. Ritzenthaler, Table of curves with many points available at http://www.manypoints.orgGoogle Scholar

[5]

M. GiuliettiM. Montanucci and G. Zini, On maximal curves that are not quotients of the Hermitian curve, Finite Fields Appl, 41 (2016), 72-88. doi: 10.1016/j.ffa.2016.05.005. Google Scholar

[6]

E. Kani and M. Rosen, Idempotent relations and factors of Jacobians, Math. Ann., 284 (1989), 307-327. doi: 10.1007/BF01442878. Google Scholar

[7]

M. Q. Kawakita, Wiman's and Edge's sextic attaining Serre's bound preprint, available at http://www.manypoints.org/upload/kawakita.pdfGoogle Scholar

[8]

M. Q. Kawakita, Wiman's and Edge's sextic attaining Serre's bound Ⅱ, in Algorithmic Arithmetic, Geometry, and Coding Theory, Amer. Math. Soc., 2015, 191–203. doi: 10.1090/conm/637/12758. Google Scholar

[9]

K. Lauter, Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields, J. Algebraic Geom., 10 (2001), 19-36. Google Scholar

[10]

C. Moreno, Algebraic Curves over Finite Fields Cambridge Univ. Press, 1991. doi: 10.1017/CBO9780511608766. Google Scholar

[11]

A. Wiman, Ueber eine einfache Gruppe von 360 ebenen Collineationen, Math. Ann., 47 (1896), 531-556. doi: 10.1007/BF01445800. Google Scholar

Table 1.  Maximal curves over $\mathbb{F}_{p^2}$ of genus $6$
5 7 11 13 17 19 23 29
W W W W W
31 37 41 43 47 53 59 61
W S W S W S W
67 71 73 79 83 89 97 101
S W W W W S W
103 107 109 113 127 131 137 139
W W E S W W S W
149 151 157 163 167 173 179 181
E S S W S W S
191 193 197 199
W W S W
5 7 11 13 17 19 23 29
W W W W W
31 37 41 43 47 53 59 61
W S W S W S W
67 71 73 79 83 89 97 101
S W W W W S W
103 107 109 113 127 131 137 139
W W E S W W S W
149 151 157 163 167 173 179 181
E S S W S W S
191 193 197 199
W W S W
Table 2.  $S$ with many points over $\mathbb{F}_p$
$\mathbb{F}_p$ $(a,b,c)$ $\#S(\mathbb{F}_p)$ old entry
$19 $ $(13,6,16) $ $56$ $[50 -68]$
$37 $ $(29,28,14) $ $98 $ $[86 -104]$
$43 $ $(2,4,2)$ $104 $ $[100-116]$
$53 $ $(51,36,1) $ $ 132 $ $[120-138 ]$
$61 $ $(42,54,17)$ $140 $ $[134-152] $
$67 $ $(65,2,45) $ $152$ $[140-164] $
$71 $ $(29,65,70)$ $ 156 $ $[150-168] $
$\mathbb{F}_p$ $(a,b,c)$ $\#S(\mathbb{F}_p)$ old entry
$19 $ $(13,6,16) $ $56$ $[50 -68]$
$37 $ $(29,28,14) $ $98 $ $[86 -104]$
$43 $ $(2,4,2)$ $104 $ $[100-116]$
$53 $ $(51,36,1) $ $ 132 $ $[120-138 ]$
$61 $ $(42,54,17)$ $140 $ $[134-152] $
$67 $ $(65,2,45) $ $152$ $[140-164] $
$71 $ $(29,65,70)$ $ 156 $ $[150-168] $
Table 3.  $S$ with many points over $\mathbb{F}_q$
$\mathbb{F}_q$ $(a,b,c)$ $\#S(\mathbb{F}_q)$ old entry
$5^3 $ $(\beta^4,\beta^{56},\beta^{38})$ $ 240 $ $[210-255] $
$u^3+3u+3=0$
$7^3 $ $(\beta^{22},\beta^{94},\beta^{8})$ $ 542 $ $[512-564] $
$u^3-u^2+4=0$
$\mathbb{F}_q$ $(a,b,c)$ $\#S(\mathbb{F}_q)$ old entry
$5^3 $ $(\beta^4,\beta^{56},\beta^{38})$ $ 240 $ $[210-255] $
$u^3+3u+3=0$
$7^3 $ $(\beta^{22},\beta^{94},\beta^{8})$ $ 542 $ $[512-564] $
$u^3-u^2+4=0$
[1]

Domokos Szász. Algebro-geometric methods for hard ball systems. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 427-443. doi: 10.3934/dcds.2008.22.427

[2]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151

[3]

Daniele Bartoli, Adnen Sboui, Leo Storme. Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes. Advances in Mathematics of Communications, 2016, 10 (2) : 355-365. doi: 10.3934/amc.2016010

[4]

Roland D. Barrolleta, Emilio Suárez-Canedo, Leo Storme, Peter Vandendriessche. On primitive constant dimension codes and a geometrical sunflower bound. Advances in Mathematics of Communications, 2017, 11 (4) : 757-765. doi: 10.3934/amc.2017055

[5]

Srimanta Bhattacharya, Sushmita Ruj, Bimal Roy. Combinatorial batch codes: A lower bound and optimal constructions. Advances in Mathematics of Communications, 2012, 6 (2) : 165-174. doi: 10.3934/amc.2012.6.165

[6]

Leonardo Manuel Cabrer, Daniele Mundici. Classifying GL$(n,\mathbb{Z})$-orbits of points and rational subspaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4723-4738. doi: 10.3934/dcds.2016005

[7]

Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583

[8]

Zihui Liu, Xiangyong Zeng. The geometric structure of relative one-weight codes. Advances in Mathematics of Communications, 2016, 10 (2) : 367-377. doi: 10.3934/amc.2016011

[9]

T. L. Alderson, K. E. Mellinger. Geometric constructions of optimal optical orthogonal codes. Advances in Mathematics of Communications, 2008, 2 (4) : 451-467. doi: 10.3934/amc.2008.2.451

[10]

J. De Beule, K. Metsch, L. Storme. Characterization results on weighted minihypers and on linear codes meeting the Griesmer bound. Advances in Mathematics of Communications, 2008, 2 (3) : 261-272. doi: 10.3934/amc.2008.2.261

[11]

Florent Foucaud, Tero Laihonen, Aline Parreau. An improved lower bound for $(1,\leq 2)$-identifying codes in the king grid. Advances in Mathematics of Communications, 2014, 8 (1) : 35-52. doi: 10.3934/amc.2014.8.35

[12]

Carlos Munuera, Fernando Torres. A note on the order bound on the minimum distance of AG codes and acute semigroups. Advances in Mathematics of Communications, 2008, 2 (2) : 175-181. doi: 10.3934/amc.2008.2.175

[13]

Yael Ben-Haim, Simon Litsyn. A new upper bound on the rate of non-binary codes. Advances in Mathematics of Communications, 2007, 1 (1) : 83-92. doi: 10.3934/amc.2007.1.83

[14]

José Joaquín Bernal, Diana H. Bueno-Carreño, Juan Jacobo Simón. Cyclic and BCH codes whose minimum distance equals their maximum BCH bound. Advances in Mathematics of Communications, 2016, 10 (2) : 459-474. doi: 10.3934/amc.2016018

[15]

Weishi Liu. Geometric approach to a singular boundary value problem with turning points. Conference Publications, 2005, 2005 (Special) : 624-633. doi: 10.3934/proc.2005.2005.624

[16]

Frédéric Vanhove. A geometric proof of the upper bound on the size of partial spreads in $H(4n+1,$q2$)$. Advances in Mathematics of Communications, 2011, 5 (2) : 157-160. doi: 10.3934/amc.2011.5.157

[17]

Vadim Yu. Kaloshin and Brian R. Hunt. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II. Electronic Research Announcements, 2001, 7: 28-36.

[18]

Vadim Yu. Kaloshin and Brian R. Hunt. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I. Electronic Research Announcements, 2001, 7: 17-27.

[19]

Seungkook Park. Coherence of sensing matrices coming from algebraic-geometric codes. Advances in Mathematics of Communications, 2016, 10 (2) : 429-436. doi: 10.3934/amc.2016016

[20]

Christine Bachoc, Gilles Zémor. Bounds for binary codes relative to pseudo-distances of $k$ points. Advances in Mathematics of Communications, 2010, 4 (4) : 547-565. doi: 10.3934/amc.2010.4.547

2018 Impact Factor: 0.879

Metrics

  • PDF downloads (10)
  • HTML views (4)
  • Cited by (1)

Other articles
by authors

[Back to Top]