Minimum codegree condition for perfect matchings in k-partite k-graphs

Let H be a k-partite k-graph with n vertices in each partition class, and let delta(k-1) (H) denote the minimum codegree of H. We characterize those H with delta(k-1) (H) >= n/2 and with no perfect matching. As a consequence, we give an affirmative answer to the following question of Rodl and Rucinski: if k is even or n not equivalent to 2 (mod 4), does delta(k-1) (H) >= n/2 imply that H has a perfect matching? We also give an example indicating that it is not sufficient to impose this degree bound on only two types of (k - 1)-sets.