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This paper provides a simplified proof for the existence of nested latticecodebooks allowing to achieve the capacity of the additive white Gaussian noisechannel, as well as the optimal rate-distortion trade-off for a Gaussiansource. The proof is self-contained and relies only on basic probabilistic andgeometrical arguments. An ensemble of nested lattices that is different, andmore elementary, than the one used in previous proofs is introduced. Thisensemble is based on lifting different subcodes of a linear code to theEuclidean space using Construction A. In addition to being simpler, ouranalysis is less sensitive to the assumption that the additive noise isGaussian. In particular, for additive ergodic noise channels it is shown thatthe achievable rates of the nested lattice coding scheme depend on the noisedistribution only via its power. Similarly, the nested lattice source codingscheme attains the same rate-distortion trade-off for all ergodic sources withthe same second moment.

A Simple Proof for the Existence of "Good" Pairs of Nested Lattices