Minimal symmetric differences of lines in projective planes

TL;DR

This paper investigates the minimal symmetric differences of lines in projective planes over finite fields, establishing bounds on the function that measures these differences for various numbers of lines.

Contribution

The paper provides new bounds on the function f(r), showing it is proportional to q for a range of r in the projective plane PG(2,q).

Findings

f(r) = O(q) for Cq^{3/2} < r < q^2 - Cq^{3/2}

Established bounds improve understanding of line differences in finite projective planes

Results apply to Desarguesian projective planes over finite fields.

Abstract

Let q be an odd prime power and let f(r) be the minimum size of the symmetric difference of r lines in the Desarguesian projective plane PG(2,q). We prove some results about the function f(r), in particular showing that there exists a constant C>0 such that f(r)=O(q) for Cq^{3/2}<r<q^2 - Cq^{3/2}.