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 Fₐ and let kq denote the family of all k-dimensional subspaces of V. A family F kq 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 kq is called a q-Steiner system Sₐ (t, k, n) if for every T tq, 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 kq. Secondly, we characterize the structures of all maximum cover-free families which are closely related to q-Steiner systems.