论文标题
关于最大弧引起的部分几何形状
On partial geometries arising from maximal arcs
论文作者
论文摘要
本文的主题是部分几何$ pg（s，t，α）$，带有参数$ s = d（d'-1），\ t = d'（d-1），\α=（d'-1）（d'-1）$，$ d，$ d，$ d，d，d'\ ge 2 $。在所有已知示例中，$ q = dd' $是2的功率，部分几何源于$ d $ $ d $ $ d $或$ d'U的最大弧线，在订单$ q $的投影平面中通过已知的构建构造，这是由于thas \ cite {thas73}而导致的{thas73}，wallis \ cite \ cite {w}，由已知的$ $ pg $ pg（4，$ pg）（4） \ cite {Math}与订单8的投影平面中与最大弧无关。平行类是一组覆盖点集的成对分离线。如果两个平行类完全共享一行，则称为正交类。证明了部分几何形状$ g $带有参数$ pg（d（d'-1），d'（d-1），（d-1），（d-1）（d'-1）$的最大成对正交平行类的最大限制，证明了$ g $的必要条件，并且是从$ d $ d $ d $ d $ d'qudive的必要条件下出现的。 $ g $及其双几何形状包含符合上限的成对正交平行类。提出了Mathon的部分几何形状的替代结构，并使用新的必要条件来证明为什么这种部分几何形状与8顺式的投影平面中的任何最大弧无关。与所有已知最大弧相关的部分最大弧相关的部分几何是16的投射平面上的所有最大弧相关的阶段，并将其分类为同位阶段，以及它们的划分和平行的划分。基于这些结果，提出了一些开放问题和猜想。
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.
