  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.
