Shaping for Construction D Lattices

In this paper we propose a practical and general method for shaping a coding lattice $\Lambda_{c}$ obtained by the construction D. The proposal assumes only that the coarse/shaping lattice $\Lambda_{s}$ is an integer lattice. To achieve this, a new way to encode and index a construction-D Voronoi lattice code is presented which has linear complexity in the block length. We test the performance of our proposal using a construction-D lattice obtained from an extended BCH code, and shaping is performed by a Gosset Lattice ($E_8$). This results in a shaping gain of 0.54 dB, which is close to the expected gain of the $E_8$, 0.65 dB.