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Finite Projective Planes and Orthogonal Arrays

有限射影平面和正交阵列

基本信息

批准号：
15540146
负责人：
SUETAKE Chihiro
金额：
$ 1.6万
依托单位：
Oita University (2005)Fukushima National College of Technology (2003-2004)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540146/
关键词：
finite projective plane automorphism group orthogonal array finite geometry incidence structure incidence matrix transversal design symmetric transversal design 有数幾何学

项目摘要

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.

我们得到了以下四个结果。(1)通过证明正交阵列OA（72,12,6,2）的不存在性，证明了12阶射影平面不存在16阶共线群。我们用电脑做研究。(2)证明了只有一种对称横向设计STD_4[12]；[3]直到同构。我们还证明了STD_4的全自同构群的序[12]；3]是2^5·3^3和Aut STD_4[12；3]在点集上有正则子群作为置换群。我们用电脑做研究。(3)构造了一个对称横截面符号STD_7[21]；3]承认一个7阶的自同构群，它半正则地作用于点群的集合和块群的集合。(4)设S为任意9阶投影平面Pi上与8点内某直线相交的块半函数。根据1999年的Dover， 2014年的leqvert Svertleq 24。在1986年，一位作者证明了如果Pi是无论证的，那么22leqvert Svertleq 24。本文给出了在所有9阶非灭参子投影平面上所有具有此性质的块半椭圆的完全确定。在任意非非论证平面Pi上，证明了21leqvert Svertleq 24和对于每一个iin{21,22,23,24}存在大小为I的阻塞半椭圆，它们与8个点上的某条直线相交。因此，多佛河的边界并不锐利。