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

In this paper we consider the diversity-multiplexing gain tradeoff (DMT) of so-called minimum delay asymmetric space-time codes for the
$n \times m$
MIMO channel. Such codes correspond to lattices in
$M_{n}(\mathbb {C})$
with dimension smaller than
$2n^{2}$
. Currently, very little is known about their DMT, except in the case
$m=1$
, corresponding to the multiple input single output (MISO) channel. 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
$\mathbb {Q}$
-central division algebras satisfy this optimality criterion. As a corollary this result provides a DMT classification for all
$\mathbb {Q}$
-central division algebra codes that are based on standard embeddings. While the
$\mathbb {Q}$
-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.