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

中文翻译：

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四个元素的域上的属 8 曲线上的最大点数

Oesterle界表明有限域$\mathbb{F}_4$上的属8曲线最多可以有24个有理点，Niederreiter和Xing用类场论证明存在这样一条21点的曲线。我们改进了这两个结果：我们证明了 $\mathbb{F}_4$ 上的 gen-8 曲线最多可以有 23 个有理点，我们提供了这样一个具有 22 个点的曲线的例子，即由下式定义的曲线两个方程 $y^2 + (x^3 + x + 1)y = x^6 + x^5 + x^4 + x^2$ 和 $z^3 = (x+1)y + x^ 2.$