Partitions of finite vector spaces over GF(2) into subspaces of dimensions 2 and s

A vector space partition of a finite vector space V over the field of q elements is a collection of subspaces whose union is all of V and whose pairwise intersections are trivial. While a number of necessary conditions have been proved for certain types of vector space partitions to exist, the problem of the existence of partitions meeting these conditions is still open. In this note, we consider vector space partitions of a finite vector space over the field GF (2) into subspaces of dimension 2 and dimension s 6= 2. While certain cases have been done previously (s = 1, s = 3, and s even), in our main theorem we build upon these general results to give a constructive proof for the existence of vector space partitions over GF (2) into subspaces of dimensions s and 2 of almost all types. In doing so, we introduce techniques that identify subsets of our vector space which can be viewed as the union of subspaces having trivial pairwise intersection in more than one way. These subsets are used to transform a given partition into another partition of a different type. This technique will also be useful in constructing further partitions of finite vector spaces.