Additivity of the two-dimensional Miller ideal

Let J(M(2)) denote the sigma-ideal associated with two-dimensional Miller forcing. We show that it is relatively consistent with ZFC that the additivity of J(M(2)) is bigger than the covering number of the ideal of the meager subsets of (omega)omega. We also show that Martin's Axiom implies that the additivity of J(M(2)) is 2(omega). Finally we prove that there are no analytic infinite maximal antichains in any finite product of B(omega)/fin.