New curves with many points over small finite fields

TL;DR

This paper employs class field theory and computational methods to discover new algebraic curves with many rational points over small finite fields, including a genus 17 curve over GF(2) with 18 points.

Contribution

It introduces a systematic search for curves with many points over small finite fields using abelian covers and computational techniques, settling the maximum points for certain cases.

Findings

Found a genus 17 curve over GF(2) with 18 points.

Discovered new curves with many rational points over small finite fields.

Performed exhaustive searches up to genus 50 for certain curve types.

Abstract

We use class field theory to search for curves with many rational points over small finite fields. By going through abelian covers of curves of small genus we find a number of new curves. In particular, we settle the question of how many points a curve over GF(2) of genus 17 can have, by finding one with 18 points. The search is aided by computer; in some cases it is exhaustive for this type of curve of genus up to 50.