Encoding and Indexing of Lattice Codes

Encoding and indexing of lattice codes is generalized from self-similar lattice codes to a broader class of lattices. If coding lattice Λc and shaping lattice Λs satisfy Λs Ď Λc, then Λc{Λs is 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 Λc and Λs both have generator matrices in triangular form, then encoding is always possible. (2) When Λc and Λs are 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 also considered. A condition for the existence of a group homomorphism between the information and C is given. The results are applicable to a variety of coding lattices, including Construction A, Construction D and LDLCs. The D4, E8 and convolutional code lattices are shown to be good choices for the shaping lattice. Thus, a lattice code C can be designed by selecting Λc and Λs separately, avoiding competing design requirements of self-similar lattice codes.