Diversity-multiplexing Gain Tradeoff: a Tool in Algebra?

TL;DR

This paper explores how information theory, specifically the diversity-multiplexing gain tradeoff, can be applied as a tool to analyze algebraic structures like division algebras, revealing surprising connections.

Contribution

It demonstrates how DMT bounds can be used to analyze the unit group of orders in division algebras, bridging algebra and information theory.

Findings

DMT bounds imply properties of algebraic structures

Information theory can inform algebraic analysis

Surprising connections between algebra and coding theory

Abstract

Since the invention of space-time coding numerous algebraic methods have been applied in code design. In particular algebraic number theory and central simple algebras have been on the forefront of the research. In this paper we are turning the table and asking whether information theory can be used as a tool in algebra. We will first derive some corollaries from diversity-multiplexing gain (DMT) bounds by Zheng and Tse and later show how these results can be used to analyze the unit group of orders of certain division algebras. The authors do not claim that the algebraic results are new, but we do find that this interesting relation between algebra and information theory is quite surprising and worth pointing out.