  In a recent work, Coronel et al. initiated the study of the relation between
the diversity-multiplexing tradeoff (DMT) performance of a multiuser
multiple-input multiple-output (MU-MIMO) lattice code and the rate of the decay
of the determinants of the code matrix as a function of the size of the signal
constellation. In this note, we state a simple general upper bound on the decay
function and study the promising code proposed by Badr and Belfiore in close
detail. We derive a lower bound to its decay function based on a classical
theorem due to Liouville. The resulting bound is applicable also to other codes
with constructions based on algebraic number theory. Further, we study an
example sequence of small determinants within the Badr-Belfiore code and derive
a tighter upper bound to its decay function. The upper bound has certain
conjectural asymptotic uncertainties, whence we also list the exact bound for
several finite data rates.
