Specifically, for low-density parity check (LDPC) codes, how do you apply Shannon's Noisy Coding Theorem to prove that not only do codes with zero-approaching maximum word error probability exist, but that a LDPC code exists with this property?
I assume this involves choosing an arbitrary rate R and constructing a LDPC code that is capacity-approaching, but I am unsure how to do this in a general (abstract) sense.
I have referenced "Design of Capacity-Approaching Irregular Low-Density Parity-Check Codes" by Richardson, Shokrollahi, and Urbanke, but their demonstration of capacity seems to rely on numerical analysis.
I believe that this only is analizable in the case of the BEC (Binary Erasure Channel) . See for example here https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-451-principles-of-digital-communication-ii-spring-2005/video-lectures/chap13.pdf 13.4.3