Plane-filling curves of small degree over finite fields

A plane curve $C$ in ${\mathbb{P}}^2$ defined over ${\mathbb{F}}_q$ is called plane-filling if $C$ contains every ${\mathbb{F}}_q$-point of ${\mathbb{P}}^2$. Homma and Kim, building on the work of Tallini, proved that the minimum degree of a smooth plane-filling curve is $q+2$. We study smooth plane-filling curves of degree $q+3$ and higher.