Lattice Erasure Codes of Low Rank with Noise Margins

Lattice codes of low rank are considered for an additive Gaussian noise channel with erasures. The objective is to minimize the error probability with no erasures subject to rate and power constraints while ensuring that the error probability under an allowable erasure pattern is bounded from above. Allowable erasure patterns considered here are those of fixed cardinality. Bounds on performance are derived, and several constructions are investigated for lattices in dimension four. It is shown that the problem can be viewed as a simultaneous ellipsoid packing problem, a generalization of the well known sphere packing problem.