On Tanner Graphs of Lattices and Codes - Information Theory, 1998. Proceedings. 1998 IEEE International Symposium on

The problem of finding a low-complexity Tanner graph for a general lattice A is studied. The problem is divided into two subproblems: 1) Finding an orthogonal sublattice A‘ of A which minimizes the complexity of the label code of the quotient group A/A’. 2) Constructing a simple Tanner graph for the label code obtained in part 1. The proposed approach for solving subproblem 2 can also be applied to any abelian group block code with arbitrary finite alphabets at different coordinates. The results of this work are useful in finding low-complexity graph representation of lattices and codes, which consequently results in efficient graph-based decoding algorithms. I. SUMMARY OF RESULTS The graph representation of codes and the corresponding decoding algorithms, initiated by the work of Gallager on lowdensity parity-check codes [2] and later generalized by Tanner [3] to codes defined by general bipartite graphs, have continued to be an active area of research particularly in the past few years (see, e.g., [SI and the references therein). The construction of Tanner graphs for linear block codes is well-known and is based on using a parity-check matrix of the code [3]. A Tanner graph construction for lattices, using a method different from the one discussed in this paper, was briefly sketched in [5]. In this work, we develop Tanner graph constructions for abelian group block codes, an important application of which is to represent the label code of a lattice [l] in a given coordinate system. Our study shows that to obtain a lowcomplexity Tanner graph for an n-dimensional (n-D) lattice A, it is reasonable to divide the problem into two subproblems: 1) finding a set of 1-D Orthogonal subspaces {Wi};’,, called graph coordinate system, which minimizes the sizes gi of the label groups Gi = Pw,(A)/Aw,, where Pw,(A) and Aw, denote the projection and cross section of A on Wi, 2) deriving a simple Tanner graph for the corresponding label code of A/A’, where A’ is the orthogonal sublattice of A in {Wi}T=l. In the following, we first address subproblem 2. To simplify the presentation of the label code, we consider its isomorphic group defined over the alphabet d = Zcl x Z, x * * * x Z, , where Z, = {0,1,. .. ,g; 1). This is denoted by G(A). The dual G’ of G is also defined over A. It consists of those codewords c* = (c:, . . . , cz) E A such that E:=’=, cfci/gi E Z, Vc = (cl,. . . , c,) E G, where multiplications and divisions are performed in the field of real numbers. It can be seen that G’ is also an abelian group with order IG’l = ldl/lGl. The label code G is therefore fully described by the following set of check equations: