Partitions of the lines in PG[formula omitted] into multifold spreads for [formula omitted

Abstract

In 2012, Momihara et al. (2012) showed that the 2-design formed by the planes (2-flats) in AG(2n,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(1,32n) 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(2n−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.