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

The DMT of Real and Quaternionic Lattice Codes and DMT Classification of Division Algebra Codes