Incidence Matrices of Subsets—A Rank Formula

Let $n\geqq k\geqq l \geqq 0$ be integers, $\mathbb{F}$ a field, and $X = \{ 1, \cdots ,n \}$. $M = M_{n,l,k} $ is an $ \begin{pmatrix} n \\ l \end{pmatrix} \times \begin{pmatrix} n \\ k \end{pmatrix}$ matrix whose rows correspond to l-subsets of X, and columns to k-subsets of X. For $L \in X^{(l)} ,K \in X^{(k)} $ the $(L,K)$ entry of M is 1 if $L \subset K$, 0 otherwise. The problem is to find the rank of M over the field $\mathbb{F}$. We solve the problem for $\mathbb{F} = \mathbb{Z}_2 $ and obtain some result on $\mathbb{F} = \mathbb{Z}_3 $. The problem originated in extremal set theory and seems to be applicable also for matroids, codes and designs.