On Fibre Products of Kummer Curves with Many Rational Points over Finite Fields


ÖZBUDAK F. , GÜLMEZ TEMÜR B., YAYLA O.

4th International Castle Meeting Coding Theory and Applications (4ICMCTA), Palmela Castle, Portugal, 15 - 18 September 2014, vol.3, pp.307-315 identifier

  • Publication Type: Conference Paper / Full Text
  • Volume: 3
  • Doi Number: 10.1007/978-3-319-17296-5_33
  • City: Palmela Castle
  • Country: Portugal
  • Page Numbers: pp.307-315

Abstract

We determined the number of rational points of fibre products of two Kummer covers over a rational point of the projective line in a recent work of F. Ozbudak and B. G. Temur (Des Codes Cryptogr 70(3): 385-404, 2014), where we also constructed explicit examples, including a record and two new entries for the current Table of Curves with Many Points (manYPoints: Table of curves with many points. http://www.manypoints.org (2014). Accessed 30 Sep 2014). Using the methods given in Ozbudak and Gulmez Temur (Des Codes Cryptogr 70(3): 385-404, 2014), we made an exhaustive computer search over F-5 and F-7 by the contributions of O. Yayla and at the end of this search we obtained 12 records and 6 new entries for the current table; in particular, we observed that the fibre product with genus 7 and 36 rational points coincides with the Ihara bound, thus we concluded that the maximum number N-7(7) of F-7-rational points among all curves of genus 7 is 36 (Ozbudak et al., Turkish J Math 37(6): 908-913, 2013). Recently, we made another exhaustive computer search over F-11. In this paper we are representing the results as three records and three new entries for the current table.