On large complete arcs: ood case

TL;DR

This paper presents a method to compute upper bounds on the size of large complete arcs in finite geometries, utilizing geometric properties and the Hasse-Weil bound for rational points over finite fields.

Contribution

It introduces a new approach for bounding the size of large complete arcs using geometric analysis and the Stoehr-Voloch method, especially for large field orders.

Findings

Derived geometrical properties of irreducible envelopes for large complete arcs

Established upper bounds on arc sizes based on field order

Applied Hasse-Weil bound to finite field rational points

Abstract

An approach for the computation of upper bounds on the size of large complete arcs is presented. We obtain in particular geometrical properties of irreducible envelopes associated to a second largest complete arc provided that the order of the underlying field is large enough. We use Stoehr-Voloch's approach to the Hasse-Weil bound for rational points of curves defined over finite fields