Bit-optimal Decoding of Codes Whose Tanner Graphs Are Trees

We introduce a group algebra formulation for bit optimal decoding of binary block codes. We use this new framework to give a simple algebraic proof that Pearl’s and Gallager’s belief propagation decoding algorithms are bit-optimal when the Tanner graph of the code is a tree. We believe that these derivations of known results give new insights into the issues of decoding on graphs from the algebraic coding theorist’s point of view.