On cover-free families of finite vector spaces

There is a large literature on cover-free families of finite sets, because of their many applications in combinatorial group testing, cryptographic and communications. This work studies the generalization of cover-free families from sets to finite vector spaces. Let V be an n-dimensional vector space over the finite field Fq and let [V k]q denote the family of all k-dimensional subspaces of V. A family F⊆ [V k]q is called cover-free if there are no three distinct subspaces F₀, F₁, F₂∈ F such that F₀≤ (F₀∩ F₁)+(F₀∩ F₂). A family H⊆ [V k]q is called a q-Steiner system Sq(t, k, n) if for every T∈ [V t]q, there is exactly one H∈ H such that T≤ H. In this paper we investigate cover-free families in the vector space V. Firstly, we determine the maximum size of a cover-free family in [V k]q. Secondly, we characterize the structures of all maximum cover-free families which are closely related to q-Steiner systems.