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.

中文翻译：

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将 PG(2n−1,s) 中的行划分为 s=3,4 的多重扩展

摘要 2012 年，Momihara 等人。(2012) 表明，由 AG ( 2 n , 3 ) 中的平面 (2-flats) 形成的 2-设计可以分解为比以前已知的分解更多的子设计。他们将稳定所得子设计的组限制为仿射一般线性群 AGL ( 1 , 3 2 n ) 及其子群，然后给出最佳分解，即只要 n 是奇数，子设计的总数就最大。他们进一步展示了一种计算由其分解产生的子设计的确切数量的方法。在本文中，将他们的问题设置转换为 PG ( 2 n − 1 , s ) 中线的划分问题，并尽可能多地将其转化为多重传播，我们将证明它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值当 s = 3 , 4 时对任何 n 进行分区。由于 Momihara 等人的技术。(2012) 计算结果的多重传播的数量不再适用于偶数 n ，还将通过在乘法字符上使用 Weil 和来提出另一种方法，并且对于 n 的某些系列，表示结果的数量多重传播作为 n 的函数，根据它们的多重性。