Lattices Over Algebraic Integers With an Application to Compute-and-Forward

Yu-Chih Huang*, Krishna R. Narayanan, Ping-Chung Wang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


In this paper, we extend Construction A of lattices to the ring of algebraic integers of a general imaginary quadratic field that may not form a principal ideal domain (PID). We show that such a construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization. As an application, we then apply the proposed lattices to the compute-and-forward paradigm with limited feedback. Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all PIDs. A novel scheme called adaptive compute-and-forward is proposed to exploit the limited feedback about the channel state by working with the best ring of imaginary quadratic integers. Simulation results show that by adaptively choosing the best ring among the considered ones according to the limited feedback, the proposed adaptive compute-and-forward provides a better performance than that provided by the conventional compute-and-forward scheme which works over Gaussian or Eisenstein integers solely.

Original languageEnglish
Pages (from-to)6863-6877
Number of pages15
JournalIEEE Transactions on Information Theory
Issue number10
StatePublished - Oct 2018
EventIEEE International Symposium on Information Theory (ISIT) - , Hong Kong
Duration: 14 Jun 201519 Jun 2015


  • Algebraic integers
  • compute-and-forward
  • lattice codes
  • physical-layer network coding

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