Lattice Codes for the Rayleigh Fading Channel

• Constructions of Zⁿ lattices over totally real number fields.
  
  Initial search for fully diverse lattices with no restriction on the shape of the lattice resulted in either complex bit labelling procedure or loss in the average energy. This motivated the search for Zⁿ lattices over totally real number fields. The first families of such lattices are presented in:

  F. Oggier, E. Bayer-Fluckiger, E. Viterbo."New algebraic constructions of rotated cubic lattice constellations for the Rayleigh fading channel", ITW 2003, Paris.

  A complete description of several families of Zⁿ lattices as well as the analysis of their performance are detailed in:

  E. Bayer-Fluckiger, F. Oggier, E. Viterbo."New algebraic constructions of rotated Zⁿ-lattice constellations for the Rayleigh fading channel", IEEE Transactions on Information Theory, vol. 50, n.4, pp. 702-714, April 2004.

• Bounds on the performance of lattice codes and constructions of complex lattices.
  
  Once the existence of fully diverse Zⁿ-lattices has been shown, the next step has been the search for lattices with optimal performance.
  An algorithm to build lattices with improved performance in small dimensions has been proposed:

  F. Oggier, E. Bayer-Fluckiger. "Best rotated cubic lattice constellations for the Rayleigh fading channel". ISIT 2003, Yokohama.

  A bound on the performance of algebraic rotated lattices is presented in:

  E. Bayer-Fluckiger, F. Oggier and E. Viterbo. "Bounds on the Performance of Rotated Lattice Constellations". ISIT 2004, Chicago.

  With respect to this bound, we show that existing codes are almost optimal in the sense that no significant coding gain can be obtained.

  E. Bayer-Fluckiger, F. Oggier, E. Viterbo. "Algebraic lattice constellations: bounds on performance", IEEE Transactions on Information Theory, vol. 52, n. 1, January 2006.

  In this work, we also investigate the existence of complex Z[i]ⁿ-lattices.


• The whole story.
  
  The framework introduced in the above work allows to give a unified point of view on the story of the lattice codes for Rayleigh fading channels, as well as an ending since bounds were derived, and existing constructions shown to be almost optimal. The main points of the story are availabe in a survey/tutorial .