  In this paper we give new constructions of Ferrer diagram rank metric codes,
which achieve the largest possible dimension. In particular, we prove several
cases of a conjecture by T. Etzion and N. Silberstein. We also establish a
sharp lower bound on the dimension of linear rank metric anticodes with a given
profile. Combining our results with the multilevel construction, we produce
examples of subspace codes with the largest known cardinality for the given
parameters.
