We develop a Tanner graph (TG) construction for an Abelian group block code L with arbitrary alphabets at different coordinates, an important application of which is the representation of the label code of a lattice. The construction is based on the modular linear constraints imposed on the code symbols 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 efficient algorithm for finding a generating set for L*. In the process, we develop a construction for lattices based on an arbitrary Abelian group block code, called generalized Construction A (GCA), and explore relationships among a group code, its GCA lattice, and their duals. We also study the problem of finding low-complexity TGs for Abelian group block codes and lattices, and derive tight lower bounds on the label-code complexity of lattices. It is shown that for many important lattices, the minimal label codes which achieve the lower bounds cannot be supported by cycle-free Tanner graphs.

Dual code, Generalized Construction A, Group codes, Lattices, Tanner graph complexity, Tanner graph construction, Tanner graphs
IEEE Transactions on Information Theory
Department of Systems and Computer Engineering

Banihashemi, A, & Kschischang, F.R. (Frank R.). (2001). Tanner graphs for group block codes and lattices: Construction and complexity. IEEE Transactions on Information Theory, 47(2), 822–834. doi:10.1109/18.910592