Encoding and Indexing of Lattice Codes
Brian M. Kurkoski · IEEE Transactions on Information Theory · 2018
Encoding and indexing of lattice codes is generalized from self-similar lattice codes to a broader class of lattices. If coding lattice Acand shaping lattice Assatisfy As⊆ Ac, then Ac/Asis a quotient group that can be used to form a (nested) lattice code 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 Acand As both have generator matrices in a triangular form that satisfies As⊆ Ac, then encoding is always possible. 2) When Acand Asare described by full generator matrices, if a solution to a linear diophantine equation exists, then encoding is possible. In addition, special cases where C is a cyclic code are considered. A condition for the existence of a group isomorphism between the information and 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 D4, E8, etc. Thus, a lattice code C can be designed by selecting Acand Asseparately, avoiding the competing design requirements of self-similar lattice codes.