Encoding and Indexing of Lattice Codes

Abstract

Encoding and indexing of lattice codes is generalized from self-similar lattice codes to a broader class of lattices. If coding lattice $\Lambda_{\mathrm c}$ and shaping lattice $\Lambda_{\mathrm s}$ satisfy $\Lambda _{\mathrm s}\subseteq \Lambda _{\mathrm c}$ , then $\Lambda _{\mathrm c}/ \Lambda _{\mathrm s}$ is a quotient group that can be used to form a (nested) lattice code $\mathcal C$ . Conway and Sloane’s method of encoding and indexing does not apply when the lattices are not self-similar. Results are provided for two classes of lattices. 1) If $\Lambda _{\mathrm c}$ and $\Lambda _{\mathrm s}$ both have generator matrices in a triangular form that satisfies $\Lambda _{\mathrm s}\subseteq \Lambda _{\mathrm c}$ , then encoding is always possible. 2) When $\Lambda _{\mathrm c}$ and $\Lambda _{\mathrm s}$ are described by full generator matrices, if a solution to a linear diophantine equation exists, then encoding is possible. In addition, special cases where $\mathcal C$ is a cyclic code are considered. A condition for the existence of a group isomorphism between the information and $\mathcal C$ is given. The results are applicable to a variety of coding lattices, including Construction A, Construction D, and low-density lattice codes. A variety of shaping lattices may be used as well, including convolutional code lattices and the direct sum of important lattices such as $D_{4}$ , $E_{8}$ , etc. Thus, a lattice code $\mathcal C$ can be designed by selecting $\Lambda _{\mathrm c}$ and $\Lambda _{\mathrm s}$ separately, avoiding the competing design requirements of self-similar lattice codes.