Encoding and Indexing of Lattice Codes

TL;DR

This paper generalizes encoding and indexing methods for lattice codes beyond self-similar lattices, enabling broader lattice code design with practical encoding solutions for various lattice classes.

Contribution

It introduces new encoding techniques for non-self-similar lattice codes, expanding applicability to multiple lattice constructions like Construction A, D, and LDLCs.

Findings

Encoding is always possible for triangular generator matrices.

Linear Diophantine solutions enable encoding for full generator matrices.

Good shaping lattices include D4, E8, and convolutional code lattices.

Abstract

Encoding and indexing of lattice codes is generalized from self-similar lattice codes to a broader class of lattices. If coding lattice $\Lambda_{\textrm{c}}$ and shaping lattice $\Lambda_{\textrm{s}}$ satisfy $\Lambda_{\textrm{s}} \subseteq \Lambda_{\textrm{c}}$, then $\Lambda_{\textrm{c}} / \Lambda_{\textrm{s}}$ 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 $\Lambda_{\textrm{c}}$ and $\Lambda_{\textrm{s}}$ both have generator matrices in triangular form, then encoding is always possible. (2) When $\Lambda_{\textrm{c}}$ and $\Lambda_{\textrm{s}}$ 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 also considered. A condition for the existence of a group homomorphism between the information and $\mathcal{C}$ is given. The results are applicable to a variety of coding lattices, including Construction A, Construction D and LDLCs. The $D_4$, $E_8$ and convolutional code lattices are shown to be good choices for the shaping lattice. Thus, a lattice code $\mathcal{C}$ can be designed by selecting $\Lambda_{\textrm{c}}$ and $\Lambda_{\textrm{s}}$ separately, avoiding competing design requirements of self-similar lattice codes.