Finite Projective Planes and Orthogonal Arrays

我们得到了以下四个结果： (1)通过证明某正交阵OA(72,12,6,2)的不存在性，证明了不存在16阶直射群的12阶射影平面。我们使用计算机进行了研究。(2)证明了直到同构时只有一种对称横向设计STD_4[12;3]。我们还证明了 STD_4[12;3] 的完全自同构群的阶数为 2^5·3^3，并且 Aut STD_4[12;3] 有一个正则子群作为点集上的置换群。我们使用计算机进行研究。(3)我们构造了一个对称的横向设计 STD_7[21;3]，它承认 7 阶自同构群，它半规则地作用于点群集合和块群集合上。(4)令 S 为 9 阶任意射影平面 Pi 中的分块半椭圆，与 8 个点中的某条线相交。根据 Dover 在 [2] 中的说法，20leqvert Svertleq 24。在 [7] 中，其中一位作者表明，如果 Pi 是 desarguesian，则 22leqvert Svertleq 24。在本注释中，所有非 desarguesian 9 阶射影平面中具有此属性的所有块半椭圆都完全确定。在任何非笛沙格平面 Pi 中，表明 21leqvert Svertleq 24 并且对于每个 iin{21,22,23,24} 都存在大小为 I 的阻塞半椭圆，它们在 8 个点处与某条线相交。因此，多佛的边界并不尖锐。

We got the following four results.(1)It was proved that there is no projective plane of order 12 admitting a collineation group of order 16 by showing the nonexistence of some orthogonal array OA (72,12,6,2). We used a computer for our research.(2)It was proved that there is only one symmetric transversal design STD_4[12;3] up to isomorphism. We also showed that the order of the full automorphism group of STD_4[12;3] is 2^5・3^3 and Aut STD_4[12;3] has a regular subgroup as a permutation group on the point set. We used a computer for our research.(3)We constructed a symmetric transversal sesign STD_7[21;3] admitting an automorphism group of order 7 which acts semiregularly on the set of the point groups and on the set of block groups.(4)Let S be a blocking semioval in arbitrary projective plane Pi of order 9 which meets some line in 8 points. According to Dover in [2], 20leqvert Svertleq 24. In [7] one of the authors showed that if Pi is desarguesian, then 22leqvert Svertleq 24. In this note all blocking semiovals with this property in all non-desarguesian projective plane of order 9 are completely determined. In any non-desarguesian plane Pi it is shown that 21leqvert Svertleq 24 and for each iin{21,22,23,24} there exist blocking semiovals of size I which meet some line in 8 points. Therefore, the Dover's bound is not sharp.