Lower and upper bounds for some generalized arcs

TL;DR

This paper investigates the size and structure of generalized arcs in finite projective planes over fields with q elements, providing bounds and geometric configurations for small q.

Contribution

It introduces bounds on the cardinality of complete generalized arcs and analyzes their geometric configurations for small q.

Findings

Derived upper and lower bounds for the size of complete generalized arcs.

Characterized geometric configurations of generalized arcs for small q.

Provided insights into the structure of these arcs in finite projective planes.

Abstract

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.