Generate a random binary matrix with given rank by using minimum number of 1s

Hi,

I am generating a random binary matrix with size ky x w with rank min(ky,w) with the code given below. However, because I am not constraining the possibility to select 0 or 1 as an element of a matrix, number of 0s and 1s are almost equal to each other. I need to create this matrix with rank min(ky,w) using minimum number of 1s. How can I achieve this, any suggestion?

k = 5;
y = 20;
w = 104;
A = randi([0 1], k*y, w);
while gfrank(A) ~= min(k*y, w)
    A = randi([0 1], k*y, w);
end

Answer 1

[Efe Berkay Yitim]

Found a way:

k = 5;
y = 20;
w = 104;
p = 0.2;
A = randsrc(k*y, w, [0,1;1-p p]);
while gfrank(A) ~= min(k*y, w)
    A = randsrc(k*y, w, [0,1;1-p p]);
end