Lower and upper bounds for some generalized arcs

Let ${\mathbb{F}}_q$ be a field with $q$ elements. In this note, we study some generalized arcs, that is, sets of ${\mathbb{F}}_q$-points in the projective plane ${\mathbb{P}}^2({\mathbb{F}}_q)$ such that no six of them are on a conic. First, we consider the geometric configurations of such generalized arcs for small values of $q$ and then we give some upper and lower bounds for the cardinality of complete generalized arcs, i.e. generalized arcs which are not contained in a bigger one.