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

TL;DR

This paper analyzes the diversity-multiplexing gain tradeoff (DMT) of asymmetric space-time codes, especially those based on real and quaternion matrices, providing bounds and classification for division algebra codes.

Contribution

It introduces new DMT upper bounds for asymmetric codes and establishes a criterion for lattice codes to achieve these bounds, including a classification of division algebra codes.

Findings

Q-central division algebra codes satisfy the optimality criterion.

Codes based on Q-central division algebras achieve the largest DMT in their class.

Most asymmetric codes fall short of the optimal DMT except in the MISO case.

Abstract

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.