A COMPUTER-SEARCH FOR FINITE PROJECTIVE-PLANES OF ORDER-9
A COMPUTER-SEARCH FOR FINITE PROJECTIVE-PLANES OF ORDER-9
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DOI：
10.1016/0012-365x(91)90280-f
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发表时间：
1991-11-17
影响因子：
0.8
通讯作者：
THIEL, L
中科院分区：
文献类型：
--
作者：
LAM, CWH;KOLESOVA, G;THIEL, L
已知有4个9阶有限射影平面。本文报告了一次计算机搜索的结果，表明这个列表是完整的。计算机搜索首先生成所有283,657个8阶非同构拉丁方。每个拉丁方给出关联矩阵的27列。另一个程序尝试将这些关联矩阵中的每一个补充到40列。其中只有21个能够这样完成，从而产生326个40列的矩阵。第三个计算机程序尝试完成其余的矩阵。326个中的1个无法完成，其余的每个都能唯一地完成到一个矩阵。然后将一个同构测试程序应用于325个完整矩阵，为每个矩阵以及它的直射变换群创建一个证书。然后将这些证书与已知平面进行比较，没有发现新的平面。作为计算机程序正确性的进一步证据，本文还表明计算机结果与利用已知平面及其相关拉丁方的信息所预期的结果是一致的。
There are four known finite projective planes of order 9. This paper reports the result of a computer search which shows that this list is complete. The computer search starts by generating all 283,657 non-isomorphic latin squares of order 8. Each latin square gives 27 columns of the incidence matrix. Another program attempts to complete each of these incidence matrices to 40 columns. Only 21 of them can be so completed, giving rise to 326 matrices of 40 columns. A third computer program attempts to complete the rest of the matrices. One of the 326 does not complete. The rest complete each to a unique matrix. An isomorphism testing program is then applied to the 325 complete matrices, creating a certificate for each matrix, as well as its collineation group. The certificates are then compared with the known planes and no new ones found. As a further evidence of the correctness of the computer programs, this paper also shows that the computer results are consistent with those expected by using information about the known planes and their associated latin squares.