Sparse 0-1 Matrices and Forbidden Hypergraphs

Abstract

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 ℤ2. For fixed integers k ≥ 4 and r ≥ 2, where k is even and gcd(k - 1,r) = 1, we prove the lower bound N(m,k,r) = Ω(mkr/2(k-1) · (ln m)1/k-1). This improves on earlier results from [14] by a factor Θ((ln m)1/k-1). Moreover, we describe a polynomial time algorithm achieving this new lower bound.