On the rank of certain incidence matrices over \(\text{GF}(2)\) (Q1304438)

Let \(V\) be a vector space of dimension \(n\) over \(GF(2)\) and \(k\) an integer satisfying \(1\leq k \leq n-2\). Let \(S_i\) be the set of all \(i\)-dimensional subspaces of \(V\). Define a \((0,1)\)-incidence matrix \(M(n,k)\) as follows: The rows are indexed by \(S_k\) and the columns are indexed by \(S_{k+2}\). The entry in row \(r\), column \(c\) indicates whether \(r\) is a subspace of \(c\) or not. The main result of this paper is a recursive upper bound for the rank of \(M(n,k)\). One defect is that there is very little in the way of motivation or historical context for the problem provided.