The supertail of a subspace partition

Improved upper bounds for partial spreads

“…Partial k-spreads have applications in the construction of orthogonal arrays and (s, r, µ)-nets 12 , see [4]. Thus, Theorem 4.3 also implies restrictions for these objects.…”

“…The improvement of Theorem 3.5, i.e. see [12,Theorem 2], is not sufficient to exclude the case of Lemma 4.1. 8 The result is also valid for k = 2r − 1, r ≥ 2, and q ∈ {2, 3}.…”

Vector space partitions of GF(2)^8

“…Let P be a vector space partition of PG(v − 1, 2), then P cannot have a supertail of one of the following types: For literature on the supertail we refer e.g. to [14,15].…”

The structure of the minimum size supertail of a subspace partition

“…If equality holds in (3), then Theorem 2 has the following interesting corollary (see [15]). Note that the crucial part of the conclusion of Corollary 3 is that the set of all points covered by the subspaces in ST is a subspace.…”

“…To the best of our knowledge, there are not many other known necessary conditions for the existence of a subspace partition P of V . Heden and Lehmann [13] derived some necessary conditions (see Lemma 10) by essentially counting in two ways tuples of the forms (H, U) and (H, W 1 , W 2 ), where H is a hyperplane of V and U, W 1 , W 2 are subspaces of P that are contained in H. Blinco et al [5] and Heden [9,11] derived some necessary conditions on the set T of subspaces of minimum dimension (called tail in [9]) of P. The concept of tail was later generalized by Heden et al [15], as we shall see below.…”