Everett W. Howe: The maximum number of points on a curve of genus eight over the field of four elements, J. Number Theory (to appear).

(A preprint version is available.)

The Oesterlé bound shows that a curve of genus 8 over the finite field F4 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 F4 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 y2 + (x3 + x + 1)y = x6 + x5 + x4 + x2 and z3 = (x+1)y + x2.