On [formula omitted]-fold partitions of finite vector spaces and duality

Abstract

Vector space partitions of an n-dimensional vector space V over a finite field are considered in Bu (1980) [5], Heden (1984) [10], and more recently in a series of papers Blinco et al. (2008) [3], El-Zanati et al. (2008, 2009) [8,9]. In this paper, we consider the generalization of a vector space partition which we call a λ-fold partition (or simply a λ-partition). In particular, for a given positive integer, λ, we define a λ-fold partition of V to be a multiset of subspaces of V such that every nonzero vector in V is contained in exactly λ subspaces in the given multiset. A λ-fold spread as defined in Hirschfeld (1998) [12] is one example of a λ-fold partition. After establishing some definitions in the introduction, we state some necessary conditions for a λ-fold partition of V to exist, then introduce some general ways to construct such partitions. We also introduce the construction of a dual λ-partition as a way of generating λ′-partitions from a given λ-partition. One application of this construction is that the dual of a vector space partition will, in general, be a λ-partition for some λ>1. In the last section, we discuss a connection between λ-partitions and some designs over finite fields.