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The number $N_9(5)$, the maximal number of $\mathbb{F}_9$-rational points oncurves over $\mathbb{F}_9$ of genus $5$ is unknown, but it is known that $32\le N_9(5)\le 35$. In this paper, we enumerate hyperelliptic curves andtrigonal curves over $\mathbb{F}_3$ which have many $\mathbb{F}_9$-rationalpoints (and $\mathbb{F}_3$-rational points), especially the maximal number of$\mathbb{F}_9$-rational points of those curves is $30$. Kudo-Harashita studiedthe nonhyperelliptic and nontrigonal case,where they found a new example ofcurves (over $\mathbb{F}_3$) of genus five which attains $32$ and proved thatthere is no example attaining more than $32$, among sextic plane curves withmild singularities. We conclude from the main results in this paper that weneed to search sextic models (i.e., nonhyperelliptic and nontrigonal) with badsingularities, in order to find a genus-five curve over $\mathbb{F}_3$ with atleast $33$ $\mathbb{F}_9$-rational points.

Genus-five hyperelliptic or trigonal curves with many rational points in characteristic three