The Maximum Number of Points on a Curve of Genus $4$ over $\mathbb{F}_8$ is $25$

http://dx.doi.org/10.4153/CJM-2003-015-7
Canad. J. Math. 55(2003), 331-352

  Published:2003-04-01
 Printed: Apr 2003

We prove that the maximum number of rational points on a smooth, geometrically irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible by combining techniques from algebraic geometry with a computer verification. The appendix shows that 26 points is not possible by examining the zeta functions.