Ah. Banihashemi et Fr. Kschischang, Tanner graphs for group block codes and lattices: Construction and complexity, IEEE INFO T, 47(2), 2001, pp. 822-834
We develop a Tanner graph (TG) construction for an Abelian group block code
L with arbitrary alphabets at different coordinates, an important applicat
ion of which is the representation of the label code of a lattice. The cons
truction is based on the modular linear constraints imposed on the code sym
bols by a set of generators for the dual code L*,
As a necessary step toward the construction of a TG for L, we devise an eff
icient algorithm for finding a generating set for L*. In the process, we de
velop a construction for lattices based on an arbitrary Abelian group block
code, called generalized Construction A (GCA), and explore relationships a
mong a group code, its GCA lattice, and their duals,
We also study the problem of finding low-complexity TGs for Abelian group b
lock codes and lattices, and derive tight lower bounds on the label-code co
mplexity of lattices. It is shown that for many important lattices, the min
imal label codes which achieve the lower bounds cannot be supported by cycl
e-free Tanner graphs.