Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra

TL;DR

This paper explores inverse determinant sums in algebraic space-time codes, linking their growth to the diversity-multiplexing trade-off and providing a classification using ergodic theory and Lie groups.

Contribution

It establishes a general framework for analyzing inverse determinant sums and connects their asymptotic behavior to the diversity-multiplexing gain trade-off.

Findings

Growth of inverse determinant sums is determined by the unit group of the algebraic code.

Classification of inverse determinant sums for well-known codes using ergodic theory.

Reveals a connection between diversity-multiplexing trade-off and point counting in Lie groups.

Abstract

This work concentrates on the study of inverse determinant sums, which arise from the union bound on the error probability, as a tool for designing and analyzing algebraic space-time block codes. A general framework to study these sums is established, and the connection between asymptotic growth of inverse determinant sums and the diversity-multiplexing gain trade-off is investigated. It is proven that the growth of the inverse determinant sum of a division algebra-based space-time code is completely determined by the growth of the unit group. This reduces the inverse determinant sum analysis to studying certain asymptotic integrals in Lie groups. Using recent methods from ergodic theory, a complete classification of the inverse determinant sums of the most well known algebraic space-time codes is provided. The approach reveals an interesting and tight relation between diversity-multiplexing gain trade-off and point counting in Lie groups.