Codes on graphs: constraint complexity of cycle-free realizations of linear codes

Cycle-free graphical realizations of linear codes generalize trellis realizations. Given a linear code C and a cycle-free graph topology, there exists a well-defined minimal realization for C on that graph in which each constraint is a linear code with a well-defined length and dimension. The constraint complexity of the realization is defined as maximum dimension of any constraint code. There exists a graph that minimizes constraint complexity in which all internal nodes have degree 3 and all interface nodes have degree 2, and which moreover can be put in the form of a "tree-structured trellis realization." The constraint complexity of a general cycle-free graph realization can be less than that of any conventional trellis realization, but not by very much. Such realizations can yield reductions in decoding complexity even when they do not reduce constraint complexity.