Chapter Summary

Chapter Summary

Key Points

• 1.

  Multilevel coding is the Imai–Hirasawa construction. Given a partition chain of length $L = \log_2 M$ on a constellation, MLC uses one binary code per level, combined through a partition-based labelling $\mu$. The construction is modular: the binary codes can be convolutional, LDPC, polar, or any future binary code.

• 2.

  The capacity rule is the chain rule in disguise. The CM capacity decomposes exactly as $C_{\rm CM} = \sum_{i=0}^{L-1} I(Y; B_i \mid B_0, \ldots, B_{i-1}) = \sum_{i=0}^{L-1} C_i$ via the chain rule of mutual information. Setting $R_i = C_i$ at every level is the unique rate allocation that achieves the full $C_{\rm CM}$ with MLC. No other allocation works.

• 3.

  Multistage decoding realises the rule. MSD decodes the $L$ levels sequentially, feeding the decoded bits from level $i$ as side information to level $i + 1$. Its complexity is linear in $L$ rather than exponential in the aggregate rate, yet it achieves the same capacity as a joint ML decoder.

• 4.

  Error propagation in MSD is benign at the capacity operating point. A history error at stage $i - 1$ costs up to a $3$ dB distance reduction at stage $i$, but the propagation bound $P_{e,i} \le P_{e,i}^{\rm genie} + p_{<i}$ shows the penalty is negligible when each stage operates near its own threshold.

• 5.

  CM $\ge$ BICM $\ge$ sum of independent levels, always. The CM capacity dominates the BICM capacity by $\sum_{i \ge 1} I(B_i; B_{<i} \mid Y) \ge 0$. Under Gray labelling the gap is empirically tiny — well under $0.5$ bit over all practical SNRs, and essentially zero for square QAM at moderate rates.

• 6.

  BICM won the wireless modem war. Despite MLC's theoretical optimality, every major wireless standard from DVB-S2 onwards chose Gray-BICM with a single adaptive-rate binary code. The reasons are operational: one code covers every modulation, rate adaptation is trivial, interleaver depth tames fading, and iterative demapping closes most of the residual gap.

• 7.

  MLC still wins on APSK and lattices. Constellations without a good Gray labelling — notably satellite APSK — retain a substantive MLC advantage. The lattice coded modulation of Ch. 4 is essentially MLC over a sub-lattice partition chain.