Search

A vector space partition P of the projective space PG(v - 1; q) is a set of subspaces in PG(v - 1; q) which partitions the set of points. We say that a vector space partition P has type ... if precisely mi of its elements have dimension i, where 1 ≤ i ≤ v - 1. Here we determine all possible types of vector space partitions in PG(7; 2). [ABSTRACT FROM AUTHOR]

Abstract A vector space partition of F q v is a collection of subspaces such that every non-zero vector is contained in a unique element. We improve a lower bound of Heden, in a subcase, on the number of elements of the smallest occurring dimension in a vector space partition. To this end, we introduce the notion of q r -divisible sets of k -subspaces in F q v. By geometric arguments we obtain non-existence results for these objects, which then imply the improved result of Heden. [ABSTRACT FROM AUTHOR]

A partial $$(k-1)$$ - spread in $${\text {PG}}(n-1,q)$$ is a collection of $$(k-1)$$ -dimensional subspaces with trivial intersection. So far, the maximum size of a partial $$(k-1)$$ -spread in $${\text {PG}}(n-1,q)$$ was known for the cases $$n\equiv 0\pmod k$$ , $$n\equiv 1\pmod k$$ , and $$n\equiv 2\pmod k$$ with the additional requirements $$q=2$$ and $$k=3$$ . We completely resolve the case $$n\equiv 2\pmod k$$ for the binary case $$q=2$$ . [ABSTRACT FROM AUTHOR]

Let V=V(kt+r,q)V=V(kt+r,q) be a vector space of dimension kt+rkt+r over the finite field with q elements. Let σq(kt+r,t)denote the minimum size of a subspace partition PP of VV in which t is the largest dimension of a subspace. We denote by ndi the number of subspaces of dimension dni that occur in P and we say is the type of P. In this paper, we show that a partition of minimum size has a unique partition type if is an even integer. We also consider the case when t+r is an odd integer, but only give partial results since this case is indeed more intricate. [ABSTRACT FROM AUTHOR]

A subspace partition Π of V = V( n, q) is a collection of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of Π. The size of Π is the number of its subspaces. Let σ( n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let ρ( n, t) denote the maximum size of a subspace partition of V in which the smallest subspace has dimension t. In this article, we determine the values of σ( n, t) and ρ( n, t) for all positive integers n and t. Furthermore, we prove that if n ≥ 2 t, then the minimum size of a maximal partial t-spread in V( n + t −1, q) is σ( n, t). [ABSTRACT FROM AUTHOR]

Abstract: A subspace partition of is a collection of subspaces of whose pairwise intersection is empty. Let denote the minimum size (i.e., minimum number of subspaces) in a subspace partition of in which the largest subspace has dimension t. In this paper, we determine the value of for . Moreover, we use the value of to find the minimum size of a maximal partial t-spread in . [Copyright &y& Elsevier]

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

Abstract: A vector space partition of a finite dimensional vector space of dimension over a finite field with elements, is a collection of subspaces with the property that every non zero vector of is contained in exactly one of these subspaces. The tail of consists of the subspaces of least dimension in , and the length of the tail is the number of subspaces in the tail. Let denote the second least dimension in . Two cases are considered: the integer does not divide respective divides . In the first case it is proved that if then and if then either or . These lower bounds are shown to be tight and the elements in the subspaces in tails of minimal length will constitute a subspace of of dimension respectively . In case divides it is shown that if then and if then . The last bound is also shown to be tight. The results considerably improve earlier found lower bounds on the length of the tail. [Copyright &y& Elsevier]