We explain a naive approach towards the problem of finding genus 3 curves
C over any given finite field
$
\mathbb{F}_q
$
\mathbb{F}_q
of odd characteristic, with a number of rational points close to the Hasse-Weil-Serre upper bound
$
q + 1 + 3\left[ {2\sqrt q } \right]
$
q + 1 + 3\left[ {2\sqrt q } \right]
. The method turns out to be successful at least in characteristic 3.
It is a pleasure to thank Hendrik Lenstra for his interest in this work, and for his remarks which led to Section 2 of this
paper.