  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 $z^3 = (x+1)y + x^2.$
