  Hermitian hulls of linear codes are interesting for theoretical and practical
reasons alike. In terms of recent application, linear codes whose hulls meet
certain conditions have been utilized as ingredients to construct
entanglement-assisted quantum error correcting codes. This family of quantum
codes is often seen as a generalization of quantum stabilizer codes.
Theoretically, compared with the Euclidean setup, the Hermitian case is much
harder to deal with. Hermitian hulls of MDS linear codes with low dimensions
have been explored, mostly from generalized Reed-Solomon codes. Characterizing
Hermitian hulls which themselves are MDS appears to be more involved and has
not been extensively studied.
  This paper introduces some tools to study linear codes whose Hermitian hulls
are MDS. Using the tools, we then propose explicit constructions of such codes.
We consider Hermitian hulls of both Reed-Solomon and non Reed-Solomon types of
linear MDS codes. We demonstrate that, given the same Hermitian hull
dimensions, the codes from our constructions have dimensions which are larger
than those in the literature.
