Decoding of LDPC codes on the erasure channel: In Chapter 6, we illustrate different decoding algorithms for LDPC codes over erasure channels, namely, iterative (IT) and maximum likelihood (ML) decoding. Decoding on the erasure channel can be strongly simplified with respect to decoding on other channels, since whenever a symbol is not erased, we know its value with full certainty. We illustrate that iterative decoding can be done by a peeling process which resolves one unknown per iteration. ML decoding can be performed efficiently by solving a sparse system of equations by variants of the Gaussian elimination algorithm.

Performance analysis for iterative decoders: In Chapter 7, we discuss the behavior of LDPC codes under iterative erasure decoding. When the blocklength goes to infinity, for many LDPC codes, the symbol error probability exhibits a so-called threshold phenomenon that is, there exists a certain channel erasure probability below which error-free communication is possible, while this is not guaranteed above it. We discuss how to compute this threshold for ensembles of LDPC codes on memoryless erasure channels. In the finite-length setting, one may observe a flattening of the symbol error rate curve owing to stopping sets – specific structures in the code’s bipartite graph. Knowing their number and size allows predicting this so-called error floor. Based on our findings, we discuss how to design good LDPC for memoryless erasure channels with extension to channels with memory.