The subject of this paper are partial geometries $pg(s,t,α)$ with parameters $s=d(d'-1), \ t=d'(d-1), \ α=(d-1)(d'-1)$, $d, d' \ge 2$. 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.
Partial flocks of the quadratic cone yielding Mathon maximal arcs
N.汉密尔顿和J. A. Thas描述了Mathon型的极大弧和二次锥的部分群之间的联系。这种联系具有相当代数的性质。在本文中，我们建立了这两个结构之间的几何连接。我们还定义了一个组合物上的羊群飞机，并使用这个工作出一个类似的合成版本的Mathon定理。最后，我们展示了如何有可能构建一个最大的弧Mathon型的程度2D，包含Denniston弧的程度d提供了一个解决方案，以一定的给定系统的跟踪条件。
