Partitions of V(n, q) into 2‐ and s‐Dimensional Subspaces

Abstract

Let V=V(n,q) denote a vector space of dimension n over the field with q elements. A set scriptP of subspaces of V is a (vector space) partition of V if every nonzero element of V is contained in exactly one subspace in scriptP. Suppose that scriptP is a partition of V with xi subspaces of dimension di for 1≤i≤k. Then we call d1x1...dkxk the type of the partition scriptP. Which possible types correspond to actual partitions is in general an open question. We prove that for any odd integer s≥3 and for any integer n≥2s, the existence of partitions of V(n,q) across a suitable range of types sx2y guarantees the existence of partitions of V(n+js,q) of essentially all the types sx2y for any integer j≥1. We then apply this result to construct new classes of partitions of V. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 467‐482, 2012

On projective $q^r$-divisible codes

“…More generally, we can conclude some non-existence results for vector space partitions, whose classification is an ongoing, very hard, major project, see e.g. [32,48,49,50,83,96]. To that end, we mention the following theorem.…”

Divisible Codes

“…Nevertheless, the mentioned classification is an ongoing, very hard, major project, see e.g. [62,92,93,94,138,165]. Currently all feasible types of vector space partitions of PG( − 1, 2) with ≤ 7 are characterized [62].…”

Reconfiguration of subspace partitions

“…x y to exist. For a = 2 and b > 3, the problem of determining the partitions of V n q ( , ) of type b 2 x y was considered by Seelinger et al in a series of two papers [17,18]. In [17], they proved that the existence of subspace partitions of V n q ( , ) of type b 2 x y for a suitable range of solutions x y ( , ) implies the existence of subspace partitions of V n b q ( + , ) of type b 2 x y for almost all solutions x y ( , ).…”

“…For a = 2 and b > 3, the problem of determining the partitions of V n q ( , ) of type b 2 x y was considered by Seelinger et al in a series of two papers [17,18]. In [17], they proved that the existence of subspace partitions of V n q ( , ) of type b 2 x y for a suitable range of solutions x y ( , ) implies the existence of subspace partitions of V n b q ( + , ) of type b 2 x y for almost all solutions x y ( , ). In their follow-up paper [18], they focused on the case q = 2 and proved the existence of partitions of V n ( , 2) of type b 2 x y for almost all solutions x y ( , ) without any precondition.…”

“…Our approach combines the methods in [17,18] with the properties of double cosets. 1 This is similar to the method of switching reguli in finite geometry.…”