  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.
