On -fold partitions of finite vector spaces and duality

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.