Shaping Low-Density Lattice Codes Using Voronoi Integers

Abstract—A lattice code construction that employs two sep-arate 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 ” Zm{Λs, the set of integers inside the fundamental Voronoi region of a shaping lattice Λs, 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 E8 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 E8 lattice, that is, as much as 0.65 dB. This shaping operation can be implemented with lower complexity than previous LDLC approaches. I