Burcu TEMUR GULMEZ, Oğuz YAYLA, Ferruh OZBUDAK

An exhaustive computer search for finding new curves with many points among fibre products of two Kummer covers over $Bbb{F}_5$ and $Bbb{F}_7$

In this paper we make an exhaustive computer search for finding new curves with many points among fibre products of 2 Kummer covers of the projective line over F5 and F7 . At the end of the search, we have 12 records and 6 new entries for the current Table of Curves with Many Points. In particular, we observe that the fibre product $y^3_1$ = $frac {5(x+2)(x +5)} {x}$, $y^3_2$ $frac {3x^2(x +5)} {x + 3}$ over F7 has genus 7 with 36 rational points. As this coincides with the Ihara bound, we conclude that the maximum number N7 (7) of F7 -rational points among all curves of genus 7 is 36. Our exhaustive search has been possible because of the methods given in the recent work by Özbudak and Temür (2012) for determining the number of rational points of such curves.

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[1] Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235–265 (1997).
[2] Garcia, A., Garzon, A.: On Kummer covers with many rational points over finite fields. J. Pure Appl. Algebra 185, 177–192 (2003).
[3] Hirschfeld, J.W.P., Korchmaros, G., Torres, F.: Algebraic Curves over a Finite Field. Princeton Series in Applied Mathematics. Princeton Univ. Press, Princeton, NJ (2008).
[4] Ihara, Y.: Some remarks on the number of rational points of algebraic curves. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28, 721–724 (1982).
[5] Kawakita, M.Q.: Kummer curves and their fibre products with many rational points. Appl. Algebra Engrg. Comm. Comput. 14, 55–64 (2003).
[6] ManyPoints Table of Curves with Many Points, http://www.manypoints.org (Accessed June 13, 2012).
[7] Niederreiter, H., Xing, C.: Rational Points on Curves over Finite Fields. Cambridge University Press. Cambridge (2001).
[8] Niederreiter, H., Xing, C.: Algebraic Geometry in Coding Theory and Cryptography. Princeton Univ. Press, Princeton, NJ (2009).
[9] Ozbudak, F., Stichtenoth, H. Curves with many points and configurations of hyperplanes over finite fields. Finite Fields Appl. 5, no. 4, 436–449 (1999).
[10] Ozbudak, F., Tem ̈r, B. G.: Fibre products of Kummer covers and curves with many rational points. Appl. Algebra Engrg. Comm. Comput. 18, 433–443 (2007).
[11] Ozbudak, F. Temür, B. G.: Finite number of fibre products of Kummer covers and curves with many rational points over finite fields. Des. Codes Cryptogr. DOI 10.1007/s10623-012-9706-2.
[12] Rökaeus, K.: Computer search for curves with many points among abelian covers of genus 2 curves. arXiv:1106.5176v1 (2011).
[13] Rökaeus, K.: New curves with many points over small finite fields. arXiv:1204.4355v1, (2012)
[14] R ̈kaeus, K.: Computer search for curves with many points among unramified covers of genus 2 curves over F5 , F7 , F9 and F11 . Report, available under manYPoints.org, http://manypoints.org/upload/1369548137.pdf
[15] R ̈kaeus, K.: Computer search for curves with many points among abelian covers of genus 2 curves over F13 . Report, available under manYPoints.org, http://manypoints.org/upload/1225383491.pdf
[16] Stichtenoth, H.: Algebraic Function Fields and Codes, Second edition. Graduate Texts in Mathematics 254. Springer-Verlag, Berlin (2009).
[17] Tsfasman, M.A. Vladut, S.G. Nogin, D.: Algebraic Geometric Codes: Basic Notions. Mathematical Surveys and Monographs 139. American Mathematical Society, Providence, RI (2007).
[18] van der Geer G., van der Vlugt M.: Tables of curves with many points. Math. Comput. 69 no.230, 797–810 (2000).
[19] van der Geer, G.: Hunting for curves with many points. In IWCC2009 (Eds.: Xing, C. et al.) Lecture Notes in Computer Science 5557, 82–96 (2009).