The maximum number of points on a curve of genus 4 over F_8 is 25
  This paper proves that the maximum number of rational points on a smooth, absolutely irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible using techniques from algebraic geometry. The appendix shows that 26 points is not possible by examining the zeta functions. 
David Savitt, Kristin Lauter