On partial geometries arising from maximal arcs

TL;DR

This paper investigates partial geometries derived from maximal arcs in projective planes, establishing bounds on orthogonal parallel classes, providing new constructions, and classifying geometries for certain orders.

Contribution

It introduces a necessary and sufficient condition for partial geometries to originate from maximal arcs, and classifies known geometries for order 16.

Findings

Upper bound on orthogonal parallel classes in partial geometries.

New construction of Mathon's partial geometry.

Classification of geometries related to maximal arcs in order 16.

Abstract

The subject of this paper are partial geometries $pg(s,t,\alpha)$ with parameters $s=d(d'-1), \ t=d'(d-1), \ \alpha=(d-1)(d'-1)$, $d, d' \ge 2$. In all known examples, $q=dd'$ is a power of 2 and the partial geometry arises from a maximal arc of degree $d$ or $d'$ in a projective plane of order $q$ via a known construction due to Thas \cite{Thas73} and Wallis \cite{W}, with a single known exception of a partial geometry $pg(4,6,3)$ found by Mathon \cite{Math} that is not associated with a maximal arc in the projective plane of order 8. A parallel class of lines is a set of pairwise disjoint lines that covers the point set. Two parallel classes are called orthogonal if they share exactly one line. An upper bound on the maximum number of pairwise orthogonal parallel classes in a partial geometry $G$ with parameters $pg(d(d'-1),d'(d-1),(d-1)(d'-1))$ is proved, and it is shown that a necessary and sufficient condition for $G$ to arise from a maximal arc of degree $d$ or $d'$ in a projective plane of order $q=dd'$ is that both $G$ and its dual geometry contain sets of pairwise orthogonal parallel classes that meet the upper bound. An alternative construction of Mathon's partial geometry is presented, and the new necessary condition is used to demonstrate why this partial geometry is not associated with any maximal arc in the projective plane of order 8. The partial geometries associated with all known maximal arcs in projective planes of order 16 are classified up to isomorphism, and their parallel classes of lines and the 2-rank of their incidence matrices are computed. Based on these results, some open problems and conjectures are formulated.