A general lower bound on the decoding complexity of sparse-graph codes

Historically, the discovery of new code families and decoding algorithms has been followed by fundamental limits on their required complexity. To arrive at such limits, it is necessary to have general models that capture the essence of the approach to decoding, while still allowing for the possibility of as yet undiscovered families of codes. By adapting sphere-packing arguments used earlier in the context of delay, we derive such general lower bounds on the complexity of approaching capacity under an idealized message-passing model for decoding. The bounds demonstrate that the number of decoding iterations must increase to infinity as a system approaches capacity and upper bounds reveal them to be orderoptimal under the idealized decoding model.