Summary: 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