Sparse 0-1 matrices and forbidden hypergraphs

We consider the problem of determining the maximum number N(m,k,r) of columns of a 0-1 matrix with m rows and exactly r ones in each column such that every k columns are linearly independent over Z(2). For fixed integers k greater than or equal to 4 and r greater than or equal to 2, where k is even and gcd(k - 1, r) = 1, we prove the lower bound N(m,k,r) = Omega(m(kr/2(k-1)) . (ln m)(1/k-1)). This improves on earlier results from [14] by a factor Theta((ln m)(1/k-1)). Moreover, we describe a polynomial time algorithm achieving this new lower bound.