Towards a complete DMT classification of division algebra codes

TL;DR

This paper extends bounds on the diversity multiplexing gain trade-off for division algebra-based lattice codes, using ergodic theory to analyze their performance across a broader multiplexing gain range.

Contribution

It introduces new bounds for these codes' trade-offs by applying ergodic theory, revealing a classification based on Hasse invariants and their impact on performance.

Findings

Codes with ramification at infinite places offer better trade-offs.

Extended bounds cover a larger multiplexing gain range.

Division algebra subclasses can be distinguished by Hasse invariants.

Abstract

This work aims at providing new bounds for the diversity multiplexing gain trade-off of a general class of division algebra based lattice codes. In the low multiplexing gain regime, some bounds were previously obtained from the high signal-to-noise ratio estimate of the union bound for the pairwise error probabilities. Here these results are extended to cover a larger range of multiplexing gains. The improvement is achieved by using ergodic theory in Lie groups to estimate the behavior of the sum arising from the union bound. In particular, the new bounds for lattice codes derived from Q-central division algebras suggest that these codes can be divided into two subclasses based on their Hasse invariants at the infinite places. Algebras with ramification at the infinite place seem to provide better diversity-multiplexing gain tradeoff.