Planar Functions and Planes of Lenz-Barlotti Class II
Planar Functions and Planes of Lenz-Barlotti Class II
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DOI：
10.1023/a:1008292303803
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发表时间：
1997-02
期刊：
影响因子：
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通讯作者：
Robert S. Coulter;R. Matthews
中科院分区：
文献类型：
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作者：
Robert S. Coulter;R. Matthews
Dembowski和Ostrom [4]引入了平面函数来描述具有特殊性质的直射群的射影平面。描述了有限域上的几类平面函数，包括一类其相关仿射平面不是平移平面或对偶平移平面的函数。这就否定了[4]中提出的一个问题。这些平面函数定义了至少一个这样的仿射平面的阶为3e的每一个e ≥ 4和他们的射影闭包的Lenz-Barlotti II型。所有以前已知的II型平面都是通过推导或提升获得的。至少当e是奇数时，这里描述的平面不能以这种方式获得。
Planar functions were introduced by Dembowski and Ostrom [4] to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3efor every e ≥ 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.