Averaging bounds for lattices and linear codes

General random coding theorems for lattices are derived from the Minkowski-Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski-Hlawka theorem itself is obtained as the limit, for p→∞, of a simple lemma for linear codes over GF(p) used with p-level amplitude modulation. The relation between the combinatorial packing of solid bodies and the information-theoretic “soft packing” with arbitrarily small, but positive, overlap is illuminated. The “soft-packing” results are new. When specialized to the additive white Gaussian noise channel, they reduce to (a version of) the de Buda-Poltyrev result that spherically shaped lattice codes and a decoder that is unaware of the shaping can achieve the rate 1/2 log₂ (P/N)