Maximum-likelihood LDPC decoder analysis. In Chapter 8, the performance of LDPC codes under ML decoding is analyzed. ML decoding is intended here either as the block-wise or the symbol-wise decoding criterion (see Section 2.2). More specifically, the asymptotic analysis on the ML decoding threshold addresses the performance in terms of symbol-wise ML decoding, whereas finite-length bounds are provided for the block error probability under block-wise ML decoding. While the focus is on unstructured LDPC code ensembles, the results in this chapter can be considered to a large extent valid for other LDPC code ensembles.

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.