Towards a complete DMT classification of division algebra codes

This work aims at providing new lower bounds for the diversity-multiplexing gain trade-off of a general class of lattice codes based on division algebras. 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 classes based on their Hasse invariants at the infinite places. Algebras with ramification at the infinite place seem to provide a better diversity-multiplexing gain trade-off.