On the length of the tail of a vector space partition

A vector space partition of a finite dimensional vector space V=V(n,q) of dimension n over a finite field with q elements, is a collection of subspaces U₁,U₂,…,U_t with the property that every non zero vector of V is contained in exactly one of these subspaces. The tail of consists of the subspaces of least dimension d₁ in , and the length n₁ of the tail is the number of subspaces in the tail. Let d₂ denote the second least dimension in .Two cases are considered: the integer q^d₂−d₁ does not divide respective divides n₁. In the first case it is proved that if 2d₁>d₂ then n₁≥q^d₁+1 and if 2d₁≤d₂ then either n₁=(q^d₂−1)/(q^d₁−1) or n₁>2q^d₂−d₁. These lower bounds are shown to be tight and the elements in the subspaces in tails of minimal length will constitute a subspace of V of dimension 2d₁ respectively d₂.In case q^d₂−d₁ divides n₁ it is shown that if d₂<2d₁ then n₁≥q^d₂−q^d₁+q^d₂−d₁ and if 2d₁≤d₂ then n₁≥q^d₂. The last bound is also shown to be tight.The results considerably improve earlier found lower bounds on the length of the tail.