On partitions of finite vector spaces of small dimensions (Q759813)

Let \(V(n,q)\) be a vector space of dimension \(n\) over \(\text{GF}(q)\) and \(T\) a set of integers \(t_ 1,t_ 2,...,t_ k\) with \(2\leq t_ 1<t_ 2<...<t_ k\). A set \(\pi\) of subspaces of \(V\) is said to be a \(T\)-partition of \(V\) if (1) every nonzero vector of \(V\) is in exactly one member of \(\pi\), (2) for any \(t\in T\) there is a \(U\in \pi\) with \(\dim (U)=t\), and (3) \(\dim(U)\in T\) for all \(U\in \pi\). The main result of the paper is the proof that \(V(2t_ k,q)\) has a \(T\)-partition, generalizing earlier results by Beutelspacher that \(V(2t_ k,q)\) has a \(T\)-partition if \(t_ 1=2\) and the author that \(V(2t_ k,q)\) has a \(T\)-partition if \(2t_ 1<t_ k\) or \(k-1\leq 2(q-1)\).