Shaping low-density lattice codes using Voronoi integers

A lattice code construction that employs two separate lattices, a high dimension lattice for coding gain and a low-dimension lattice for shaping gain, is described. Systematic lattice encoding is a method to encode an integer sequence to a lattice point that is nearby that integer sequence. We describe the "Voronoi integers" ℤ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> /Λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> , the set of integers inside the fundamental Voronoi region of a shaping lattice Λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> , and a concrete scheme to label these integers. By first shaping the information using the Voronoi integers in low dimension, and then performing systematic lattice encoding using a high-dimension lattice, good shaping and coding gains can be simultaneously obtained. We concentrate on the case of using the E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sub> lattice for shaping and low-density lattice codes (LDLC) with dimension ~ 10,000 for coding. While optimal shaping provides a well-known 1.53 dB gain, previously reported shaping gains with LDLC lattices are on the order of 0.4 dB. The proposed method preserves the shaping gain of the E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sub> lattice, that is, as much as 0.65 dB. This shaping operation can be implemented with lower complexity than previous LDLC approaches.