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

Certain sextics with many rational points

Motoko Qiu Kawakita

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.

Keywords: Algebro-geometric codes, rational points, Serre bound.
Mathematics Subject Classification: Primary: 11G20, 14G05; Secondary: 14G50.
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. Garcia, G. 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.org Google Scholar

[5]

M. Giulietti, M. 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.pdf Google 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. Garcia, G. 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.org Google Scholar

[5]

M. Giulietti, M. 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.pdf Google 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
Table Options

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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] $
Table Options

Download as excel

$\mathbb{F}_p$ $(a,b,c)$ $\#S(\mathbb{F}_p)$ 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] $
$u^3-u^2+4=0$
Table Options

Download as excel

$\mathbb{F}_q$ $(a,b,c)$ $\#S(\mathbb{F}_q)$ 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] $
$u^3-u^2+4=0$
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