素数阶射影平面的一种新构造  

New construction of projective planes of prime order

作　　者：陈尚弟[1] 卫慧慧 

机构地区：[1]中国民航大学理学院,天津300300

出　　处：《中国民航大学学报》2014年第5期61-64,共4页Journal of Civil Aviation University of China

基　　金：国家自然科学基金项目(61179026)

摘　　要：射影平面是由欧氏平面加上一条非固有直线构成的,它在射影几何中的存在性是一个重要的研究课题。在组合设计中,射影平面与仿射平面有着密切的联系,并且一个q阶射影平面对应一个(q2+q+1,q+1,1)对称设计,一个q阶仿射平面对应一个(q2,q,1)可分解设计。本研究从无穷远元素和射影直线入手,给出射影平面的定义,进而利用矩阵的初等变换及矩阵对角线上元素的位置变换的理论,构造了一个可分解设计的平行类,最后通过增加q+1个无穷远点的方式得到素数阶射影平面的一种新构造,并且举例验证了构造的有效性和正确性。The existence of projective plane which consists of Euclidean plane and an extrinsic straight line is an important research subject in projective geometry. There is a close connection between affine plane and projective plane in combinatorial design. A projective plane of order q corresponds to a （qZ＋q＋1 ,q＋1, 1 ） symmetrical design, and an affine plane of order q corresponds to a（q2,q, 1 ） resolvable design. Beginning with infinity element and pro- jective line, the definition of projective plane is given. Furthermore, the elementary transformation of matrix and the position transformation of the diagonal elements are used to get the parallel classes of a resolvable design. Fi- nally, a new construction of projective plane of prime order is obtained by adding q＋l points into each parallel class. The correctness and efficiency of the construction are validated through an example.

关 键 词：射影平面 仿射平面 设计 矩阵 

分 类 号：O157[理学—数学]