On maximal plane curves of degree $3$ over ${\mathbb{F}}_4$,and Sziklai's example of degree $q-1$ over ${\mathbb{F}}_q$

The classification of maximal plane curves of degree $3$ over ${\mathbb{F}}_4$ will be given, which complements Hirschfeld-Storme-Thas-Voloch’s theorem on a characterization of Hermitian curves in ${\mathbb{P}}^2$. This complementary part should be understood as the classification of Sziklai’s example of maximal plane curves of degree $q-1$ over ${\mathbb{F}}_q$. Although two maximal plane curves of degree $3$ over ${\mathbb{F}}_4$ up to projective equivalence over ${\mathbb{F}}_4$ appear, they are birationally equivalent over ${\mathbb{F}}_4$ each other.