On the decay of the determinants of multiuser MIMO lattice codes

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 (MUMIMO) 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 but general upper bound on the decay function and study the promising code proposed by Badr & 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.