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 A
_c
and shaping lattice A
_s
satisfy A
_s
⊆ A
_c
, then A
_c
/A
_s
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 A
_c
and As both have generator matrices in a triangular form that satisfies A
_s
⊆ A
_c
, then encoding is always possible. 2) When A
_c
and A
_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 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 D
₄
, E
₈
, etc. Thus, a lattice code C can be designed by selecting A
_c
and A
_s
separately, avoiding the competing design requirements of self-similar lattice codes.