The supertail of a subspace partition

Abstract

Let V = V(n, q) be a vector space of dimension n over the finite field with q elements, and let d ₁ < d ₂ < ... < d _m be the dimensions that occur in a subspace partition ${\mathcal{P}}$ of V. Let σ _q (n, t) denote the minimum size of a subspace partition ${\mathcal P}$ of V, in which t is the largest dimension of a subspace. For any integer s, with 1 < s ≤ m, the set of subspaces in ${\mathcal{P}}$ of dimension less than d _s is called the s-supertail of ${\mathcal{P}}$ . The main result is that the number of spaces in an s-supertail is at least σ _q (d _s , d _s−1).