Chapter 3: Multilevel Coding and Multistage Decoding

Learning Objectives

• State the Imai–Hirasawa multilevel coding (MLC) construction: a separate binary code per level of a set-partitioning chain, concatenated with a constellation mapper
• Prove the capacity rule $C = \sum_{i=0}^{L-1} C_i$ from the chain rule of mutual information and use it to allocate rates across levels
• Describe multistage decoding (MSD) as a genie-free sequential procedure that realises the capacity rule, and analyse its error propagation
• Compare the CM, MLC/MSD, and BICM capacities and explain when MLC strictly outperforms BICM (non-Gray labelings, strongly partitioned constellations)
• Weigh the practical complexity of MLC against BICM and understand why BICM dominates modern wireless standards
• Place the construction in its historical arc: Imai–Hirasawa 1977 → Wachsmann–Fischer–Huber 1999 → DVB-S2 bit-interleaved LDPC