The Oesterlé bound shows that a genus-8 curve over the finite field **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 **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}.