A Survey of the Different Types of Vector Space Partitions

Abstract

A vector space partition is here a collection P of subspaces of a finite vector space V (n, q), of dimension n over a finite field with q elements, with the property that every non zero vector is contained in a unique member of P. Vector space partitions relates to finite projective planes, design theory and error correcting codes.In the first part of the talk I will discuss some relations between vector space partitions and other branches of mathematics. The other part of the talk contains a survey of known results on the type of a vector space partition, more precisely: the theorem of Beutelspacher and Heden on T-partitions, rather recent results of El-Zanati et al. on the different types that appear in the spaces V (n, 2), for n ≤ 8, a result of Heden and Lehmann on vector space partitions and maximal partial spreads including their new necessary condition for the existence of a vector space partition, and furthermore, I will give a theorem of Heden on the length of the tail of a vector space partition.Finally, I will also give a few historical remarks.

The maximum size of a partial spread in a finite projective space

“…A maximal partial (t−1)spread is a set of pairwise disjoint (t − 1)-dimensional subspaces which cannot be extended to a larger set. This problem has been extensively studied [10,19,21,27]. Besides their traditional relevance to Galois geometry, partial (t−1)-spreads are used to build byte-correcting codes (e.g., see [12,25]), 1-perfect mixed error-correcting codes (e.g., see [24,25]), orthogonal arrays and (s, k, λ)-nets (e.g., see [8]).…”

Partial Spreads and Vector Space Partitions

“…was not used at all in the proof. 29 By this we mean that ∆ is the largest divisor in the sense of Definition 1 or Theorem 7. 30 If t = r and r ∈ {0, 1, .…”

“…[53], is also not treated here. Further, there is a steady stream of literature that characterizes the existing types of vector space partitions in F v 2 for small dimensions v. Here, we touch only briefly on some results that are independent of the ambient space dimension v and refer to [29] otherwise.…”

Weighted external difference families and R-optimal AMD codes

“…These are known as vector space partitions, and have been extensively studied. (See [9] for a survey on vector space partitions.) Every elementary abelian p-group of order at least p 2 , for p prime has at least one non-trivial * -partition, as the following well-known construction demonstrates: Construction 4.20.…”