Codes on Graphs: Generalized State Realizations
نویسنده
چکیده
A class of generalized state realizations of codes is introduced. In the graph of such a realization, leaf edges represent symbols, ordinary edges represent states, and vertices represent local constraints on incident edges. Such a graph can be decoded by any version of the sum-product algorithm. Any factor graph representation of a code can be put into this form, and any generalized state realization can be converted to a “normalized” factor graph, without essential change in decoding complexity. Group codes are generated by group generalized state realizations. The dual of such a realization, appropriately defined, generates the dual group code. The dual realization uses the same symbol and state variables in the same graph topology as the primal realization, but replaces primal local constraints by their duals. A group code may be decoded using the dual graph, with appropriate Fourier transforms of the inputs and outputs; this can simplify decoding of high-rate codes. Examples are given of dual kernel and image representations, and (in an appendix) of dual encoder and syndrome-former realizations for dual codes.
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