Codes on graphs: normal realizations

Wiberg et al. (see European Transactions on Telelecommunications, vol.6, p.513-25, Sept./Oct. 1995) proposed graphical code realizations using three kinds of elements: symbol variables, state variables and local constraints. We focus on normal realizations, namely Wiberg-type realizations in which all symbol variables have degree 1 and state variables have degree 2. A natural graphical model of a normal realization represents states by leaf edges, states by ordinary edges, and local constraints by vertices. Any such graph may be decoded by message-passing (the sum-product algorithm). We show that any Wiberg-type realization may be put into normal form without essential change in its graph or its decoding complexity. Group or linear codes are realized by group or linear realizations. We show that an appropriately defined dual of a group or linear normal realization realizes the dual group or linear code. The symbol variables, state variables and graph topology of the dual realization are unchanged, while local constraints are replaced by their duals.