  In this paper we consider the diversity-multiplexing gain tradeoff (DMT) of
so-called minimum delay asymmetric space-time codes. Such codes are less than
full dimensional lattices in their natural ambient space. Apart from the
multiple input single output (MISO) channel there exist very few methods to
analyze the DMT of such codes. Further, apart from the MISO case, no DMT
optimal asymmetric codes are known. We first discuss previous criteria used to
analyze the DMT of space-time codes and comment on why these methods fail when
applied to asymmetric codes. We then consider two special classes of asymmetric
codes where the code-words are restricted to either real or quaternion
matrices. We prove two separate diversity-multiplexing gain trade-off (DMT)
upper bounds for such codes and provide a criterion for a lattice code to
achieve these upper bounds. We also show that lattice codes based on Q-central
division algebras satisfy this optimality criterion. As a corollary this result
provides a DMT classification for all Q-central division algebra codes that are
based on standard embeddings. While the Q-central division algebra based codes
achieve the largest possible DMT of a code restricted to either real or
quaternion space, they still fall short of the optimal DMT apart from the MISO
case.
