Isolation number versus Boolean rank

Let B be the binary Boolean algebra. The Boolean rank, or factorization rank, of a matrix A in Mm,n(B) is the smallest k such that A can be factored as an m×k times a k×n matrix. The isolation number of a matrix, A, is the largest number of entries equal to 1 in the matrix such that no two ones are in the same row, no two ones are in the same column, and no two ones are in a submatrix of A of the form . It is known that the isolation number of A is always at most the Boolean rank. This paper investigates for each k, if the isolation number of A is k what are some of the possible values of the Boolean rank of A.