Directed strongly regular graphs and matrices with low rank

For a given number r we consider the set of all values of the ratio k/n for which there exist an n by n {0,1} matrix with exactly k 1's in each row (column) and rank r. We prove that this set is finite and consider some further properties of the set. A directed strongly regular graph with parameters (n,k,\mu,\lambda,t) is a regular directed graph with degree k so that the number of paths of length two from x to y is t if x=y, \lambda if y is an outneighbour of x and \mu otherwise. An (undirected) strongly regular graph is a directed strongly regular graph with t=k. The rank of the adjacency matrix can be computed from the parameters. For a fixed rank r there are parameter sets that satisfies all other known restrictions for infinitely many values of k/n The above matrix result excludes all but finitely many of those parameter sets.