The maximum number of points on a curve of genus eight over the field of four elements

The Oesterl'e bound shows that a curve of genus 8 over the finite field ${\mathbb{F}}_4$ can have at most 24 rational points, and Niederreiter and Xing used class field theory to show that there exists such a curve with 21 points. We improve both of these results: We show that a genus-8 curve over ${\mathbb{F}}_4$ can have at most 23 rational points, and we provide an example of such a curve with 22 points, namely the curve defined by the two equations $y^2+(x^3+x+1)y=x^6+x^5+x^4+x^2$ and $z^3=(x+1)y+x^2\mathrm{.}$