On the length of the tail of a vector space partition

AbstractA vector space partition P 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,…,Ut with the property that every non zero vector of V is contained in exactly one of these subspaces. The tail of P consists of the subspaces of least dimension d1 in P, and the length n1 of the tail is the number of subspaces in the tail. Let d2 denote the second least dimension in P.Two cases are considered: the integer qd2−d1 does not divide respective divides n1. In the first case it is proved that if 2d1>d2 then n1≥qd1+1 and if 2d1≤d2 then either n1=(qd2−1)/(qd1−1) or n1>2qd2−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 qd2−d1 divides n1 it is shown that if d2