Partitions of the lines in PG(2n-1,s) into multifold spreads for s=3, 4
Partitions of the lines in PG(2n-1,s) into multifold spreads for s=3, 4
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将 PG(2n-1,s) 中的行划分为多重分布（s=3, 4）
DOI：
10.1016/j.disc.2020.111867
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发表时间：
2020
影响因子：
0.8
通讯作者：
N. Miyamoto and M. Jimbo
中科院分区：
文献类型：
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作者：
M. Mishima;N. Miyamoto and M. Jimbo
摘要2012年，Momihara et al.（2012）表明，由AG（2 n，3）中的平面（2-flats）形成的2-设计可以分解为比先前已知的分解更多的子设计。他们将稳定子设计的群限制在仿射一般线性群AGL（1，3 2 n）及其子群上，然后给出了最佳分解，即只要n是奇数，子设计的总数就最大。他们进一步演示了一种方法来计算分解产生的子设计的确切数量。在这篇文章中，将他们的问题设置为PG（2 n− 1，s）中的线的划分问题转化为尽可能多的多重分布，我们将证明他们的限制是不必要的，并提供一种方法来获得理论上的最大划分。自从Momihara et al.（2012）的方法不再适用于偶数n，另一种方法也将通过在乘法特征上使用Weil和来呈现，并且对于n的某个系列，根据其多重性将所得多重传播的数量表示为n的函数。
Abstract In 2012, Momihara et al.(2012) showed that the 2-design formed by the planes (2-flats) in AG (2 n, 3) can be decomposed into more subdesigns than a previously known decomposition. They restricted the group stabilizing the resulting subdesigns to the affine general linear group AGL (1, 3 2 n) and its subgroups, and then gave the best decomposition in the sense that the total number of subdesigns is maximum as long as n is odd. They further demonstrated a way to count the exact number of the subdesigns resulting from their decomposition. In this article, translating their problem setting as a partition problem of the lines in PG (2 n− 1, s) into as many multifold spreads as possible, we will show that their restriction is not necessary and provide a way to get theoretically maximum partition for any n when s= 3, 4. Since the technique in Momihara et al.(2012) for counting the number of the resulting multifold spreads is no longer applied for even n, another approach will be also presented through the use of Weil sums on a multiplicative character and, for some series of n, express the numbers of the resulting multifold spreads as functions of n according to their multiplicities.