Global sections of twisted ideal sheaves of distinct sheaves

Question

Let $D$ be set of $n$ distinct points in projective space $\mathbb{P}^N$, over field $\mathbb{C}$. Then I would like to know $h^0(I_D(m))$. This is just ask that number of linear independent degree $m$ polynomials vanishing on $D$. When $m$ is very large, this should have codimension $n$ in $h^0(O(m))$. My question is how small can $m$ be?