Everett W. Howe: Curves of medium genus with many points, Finite Fields Appl. 47 (2017) 145–160.

Official version here, preprint version at arXiv:1609.01838 [math.NT].

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.