Number field lattices achieve Gaussian and Rayleigh channel capacity   within a constant gap
  This paper proves that a family of number field lattice codes simultaneously achieves a constant gap to capacity in Rayleigh fast fading and Gaussian channels.   The key property in the proof is the existence of infinite towers of Hilbert class fields with bounded root discriminant. The gap to capacity of the proposed families is determined by the root discriminant.   The comparison between the Gaussian and fading case reveals that in Rayleigh fading channels the normalized minimum product distance plays an analogous role to the Hermite invariant in Gaussian channels. 
Roope Vehkalahti and Laura Luzzi
cs.IT math.IT math.NT