Bounds on the performance of belief propagation decoding

We consider Gallager's (1963) soft decoding (belief propagation) algorithm for decoding low density parity check (LDPC) codes, when applied to an arbitrary binary-input symmetric-output channel. By considering the expected values of the messages, we derive both lower and upper bounds on the performance of the algorithm. We also derive various properties of the decoding algorithm, such as robustness to the initial values of the messages. Our results apply both to regular and irregular LDPC codes.