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arXiv · 2004.08007

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

Everett W. Howe
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Abstract

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

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Everett W. Howe. 2020-04-17. The maximum number of points on a curve of genus eight over the field of four elements. https://arxiv.org/abs/2004.08007

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