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 [28] and Wallis [34], with a single known exception of a partial geometry pg(4, 6, 3) found by Mathon [22] 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.

本文的主题是参数为（s = d（d'-1），\ t = d'（d-1），\\ alpha =的部分几何\（pg（s，t，\ alpha）\）（d-1）（d'-1）\），\（d，d'\ ge 2 \）。在所有已知示例中，\（q = dd'\）是2的幂，并且部分几何形状由于已知的结构而在q阶投影平面中由度d或\（d'\）的最大弧产生，原因是Thas [28]和Wallis [34]，除了部分几何pg的一个已知例外Mathon [22]发现的（4、6、3）与投影阶数为8的最大弧不相关。平行线类别是覆盖点集的成对不相交线的集合。如果两个并行类仅共享一条线，则它们称为正交。参数\（pg（d（d'-1），d'（d-1），（d-1）（d'-1））的局部几何G中成对正交平行类的最大数目的上限）\）被证明，并且表明在阶\（q = dd'\）的投影平面上，由度数d或\（d'\）的最大弧产生的G的充要条件是都G它的对偶几何包含满足上限的成对正交平行类集。给出了Mathon部分几何的另一种构造，并使用了新的必要条件来说明为什么该部分几何不与8阶投影平面中的任何最大弧相关。将第16阶中的第1阶和第2阶分类为同构，并计算它们的平行线类别和其入射矩阵的2秩。基于这些结果，提出了一些未解决的问题和猜想。