Affine and projective planes
Affine and projective planes
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DOI：
10.1016/0012-365x(90)90003-z
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发表时间：
1990-08
期刊：
影响因子：
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通讯作者：
E. F. Assmus;J. D. Key
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文献类型：
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作者：
E. F. Assmus;J. D. Key
这项工作的目的是为有限射影平面的讨论和分类提出一个设置。过去，已经提出了两种分类方案：Lenz-Barlotti 分类（参见[8, 93.11）和 Ostrom 提出的更严格的平移平面分类（参见[19]）。我们的方法与这两者中的任何一个都相当不同，并且在很大程度上依赖于代数编码理论的结果和技术，特别是菲利普·德尔萨特的工作。该方法是通过其各个仿射部分来研究有限射影平面 17 ，为此，我们引入仿射平面 n 的外壳的概念：外壳原来是在适当的有限域 Fp 上由重合的那些对行的所有差异生成的代码表示 n 条平行线的矩阵。在所有已知的情况下，p 将是仿射平面 Ed 的阶 p’ 中涉及的素数；一般来说，它是平面的 n 阶的任意素数。令 H 为这样的船体，我们有 H c F $，正交 HL（在 F 中通常的内积中；‘）是 Fp 上的线性 (n’, k) 代码，具有最小权重 n，并且在其最小权重向量中可以找到开始的仿射平面 n。但是，在 H’= B 的最小权重向量中可以（并且经常存在）找到其他仿射平面。例如：如果 JG= AG,(4)，则 4 阶笛沙格仿射平面，B 是 (16, 11) 扩展二进制汉明码；后一个代码在其 140 个权重 4 向量中包含 112 个 AG,(4) 副本。我们研究两个核心问题：(1) 仿射平面的外壳在多大程度上决定平面？(2) 最小权重 n 的 Fp 上的各种 (n', k) 代码如何帮助对仿射（以及射影）平面进行分类？为此，我们引入“线性等价”的概念：两个仿射
The aim of this work is to suggest a setting for the discussion and classification of finite projective planes. In the past, two classification schemes have been put forward: the Lenz-Barlotti Classification (see [8, 93.11) and a more restricted classification of translation planes proposed by Ostrom (see [19]). Our approach is rather different from either of these two and rests heavily on the results and techniques of algebraic coding theory and, in particular, on the work of Philippe Delsarte.The approach is to study a finite projective plane 17 via its various affine parts and, to this end, we introduce the notion of the hull of an affine plane n: the hull turns out to be the code generated, over an appropriate finite field Fp, by all differences of those pairs of rows of an incidence matrix that represent parallel lines of n. In all known cases p will be the prime involved in the order, p’, of the affine plane Ed; in general it is any prime dividing the order n of the plane. Letting H be such a hull, we have H c F $, and the orthogonal HL (in the usual inner product in F;‘) is a linear (n’, k) code over Fp with minimum weight n and amongst its minimal-weight vectors one finds the affine plane n one began with. But, there can be (and frequently are) other affine planes to be found amongst the minimal-weight vectors of H’= B. For example: if JG= AG,(4), the desarguesian affine plane of order 4, B is the (16, 11) extended binary Hamming code; this latter code contains 112 copies of AG,(4) amongst its 140 weight-4 vectors. We investigate two central questions:(1) To what extent does the hull of an affine plane determine the plane?(2) How can the various (n’, k) codes over Fp of minimum weight n help to classify affine (and hence projective) planes? Toward this end we introduce a notion of “linear equivalence”: two affine