On the length of the tail of a vector space partition

Abstract

A 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 U 1 , U 2 , … , U t 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 d 1 in P , and the length n 1 of the tail is the number of subspaces in the tail. Let d 2 denote the second least dimension in P . Two cases are considered: the integer q d 2 − d 1 does not divide respective divides n 1 . In the first case it is proved that if 2 d 1 > d 2 then n 1 ≥ q d 1 + 1 and if 2 d 1 ≤ d 2 then either n 1 = ( q d 2 − 1 ) / ( q d 1 − 1 ) or n 1 > 2 q d 2 − d 1 . 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 2 d 1 respectively d 2 . In case q d 2 − d 1 divides n 1 it is shown that if d 2 < 2 d 1 then n 1 ≥ q d 2 − q d 1 + q d 2 − d 1 and if 2 d 1 ≤ d 2 then n 1 ≥ q d 2 . The last bound is also shown to be tight. The results considerably improve earlier found lower bounds on the length of the tail.