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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 acharacterization of Hermitian curves in $\mathbb{P}^2$. This complementary partshould be understood as the classification of Sziklai's example of maximalplane curves of degree $q-1$ over $\mathbb{F}_q$. Although two maximal planecurves 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.

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