Codes defined on graphs

Low-density parity-check codes, turbo codes, and indeed most practically decodable capacity-approaching error correcting codes can all be understood as codes defined on graphs. Graphs not only describe the codes, but, more important, they structure the operation of the sum-product decoding algorithm (or one of many possible variations), which can be used for iterative decoding. Such coding schemes have the potential to approach channel capacity, while maintaining reasonable decoding complexity. In this tutorial article we review factor graphs, which can be used to describe codes and the joint probability distributions that must be dealt with in decoding. We also review the sum-product algorithm, and show how this algorithm leads to iterative decoding algorithms for codes defined on graphs.