Curves of medium genus with many points

TL;DR

This paper develops algorithms to construct algebraic curves of genus 5, 6, and 7 with many rational points, aiming to improve the database of curves with high point counts over finite fields.

Contribution

It introduces new algorithms for constructing low-defect curves of specific genera to enhance the database of curves with many points.

Findings

Algorithms successfully produce curves with small defect for genera 5, 6, and 7.

Enhanced database of curves with many points over finite fields.

Potential applications in coding theory and cryptography.

Abstract

The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre upper bound for the number of points on the curve. We present algorithms for constructing curves of genus 5, 6, and 7 with small defect. Our aim is to be able to produce, in a reasonable amount of time, curves that can be used to populate the online table of curves with many points found at http://www.manypoints.org.