Curves of medium genus with many points

- We study curves of low genus over finite fields with many rational points.
- The defect of a curve is the Weil-Serre bound, minus the curve's number of points.
- We present algorithms for constructing curves of genus 5, 6, and 7 with small defect.
- We implemented our algorithms, and found many record-breaking curves.

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 manypoints.org.