A simple proof for the existence of 'good' pairs of nested lattices

Or Ordentlich, Uri Erez

Research output: Contribution to journalArticlepeer-review

41 Scopus citations


This paper provides a simplified proof for the existence of nested lattice codebooks allowing to achieve the capacity of the additive white Gaussian noise channel, as well as the optimal rate-distortion tradeoff for a Gaussian source. The proof is self-contained and relies only on basic probabilistic and geometrical arguments. An ensemble of nested lattices that is different, and more elementary, than the one used in the previous proofs is introduced. This ensemble is based on lifting different subcodes of a linear code to the Euclidean space using Construction A. In addition to being simpler, the analysis is less sensitive to the assumption that the additive noise is Gaussian. In particular, for additive ergodic noise channels, it is shown that the achievable rates of the nested lattice coding scheme depend on the noise distribution only via its power. Similarly, the nested lattice source coding scheme attains the same rate-distortion tradeoff for all ergodic sources with the same second moment.

Original languageEnglish
Article number7476874
Pages (from-to)4439-4453
Number of pages15
JournalIEEE Transactions on Information Theory
Issue number8
StatePublished - Aug 2016


FundersFunder number
Israel Science Foundation1956/15


    • Lattice codes
    • additive noise channels
    • linear codes


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