Chapter Summary

Chapter Summary

Key Points

• 1.

  A lattice $\Lambda \subset \mathbb{R}^n$ is a discrete additive subgroup of rank $n$. Its fundamental volume $V(\Lambda) = |\det \mathbf{G}|$ measures point density, its Voronoi region measures local error tolerance, and its kissing number $K(\Lambda)$ controls the error-floor multiplicity. Four canonical lattices carry most of the load: $\mathbb{Z}^n$, $D_n$, $E_8$, and the Leech lattice $\Lambda_{24}$.

• 2.

  A lattice partition $\Lambda / \Lambda'$ has index $|\Lambda/\Lambda'| = V(\Lambda')/V(\Lambda)$. A chain $\Lambda_0 \supset \Lambda_1 \supset \cdots \supset \Lambda_L$ of binary partitions provides $L$ label bits per lattice point, each bit selecting a coset at one level of the chain.

• 3.

  Forney's coset code $C(\Lambda/\Lambda'; \mathcal{C})$ combines a binary code $\mathcal{C}$ with a partition to select cosets. Its minimum squared Euclidean distance is $\min\!\bigl(d^2(\Lambda'), d_H \cdot d^2(\Lambda/\Lambda')\bigr)$, balancing sublattice distance against the code's Hamming distance. TCM (Ch. 2) is a coset code, MLC (Ch. 3) is a coset code, and most practical coded-modulation schemes are coset codes.

• 4.

  Coding and shaping contribute orthogonally to the gap to capacity. The total gap decomposes (in dB) as coding gain $\gamma_c$ plus shaping gain $\gamma_s$ plus residual. Coding gain depends on the code and the lattice; shaping gain depends on the constellation boundary. They cannot substitute for each other — a capacity- achieving code on uniform QAM is still $\pi e / 6 \approx 1.53$ dB short of Shannon.

• 5.

  Shaping gain equals $1 / (12 G(\Lambda_s))$. Where $G(\Lambda_s)$ is the normalised second moment of the shaping lattice. The cube $\mathbb{Z}^n$ has $G = 1/12$ (zero shaping gain); $E_8$ has $G \approx 0.072$ ($0.65$ dB); $\Lambda_{24}$ has $G \approx 0.066$ ($1.03$ dB); the $n$-ball approaches $G_\infty = 1/(2\pi e)$, giving the ultimate $1.53$ dB.

• 6.

  The $\pi e / 6$ ceiling is the Gaussian max-entropy bound in dB. A uniform distribution on a bounded region cannot have more differential entropy than a Gaussian at the same variance. Equating the two entropies gives $G \ge 1/(2\pi e)$, hence $\gamma_s \le \pi e / 6$. The proof is four lines; the bound is universal across all finite-alphabet schemes.

• 7.

  Practical shaping trades complexity for gain. Shell mapping (Laroia–Farvardin–Tretter, 1994, used in V.34) implements shaping via an arithmetic coder over energy shells in a 16-D base alphabet; it achieves $\approx 0.8$ dB. Trellis shaping (Forney, 1992) uses a Viterbi search over coset representatives; it achieves $\approx 1$ dB. Both pay a modest rate-adaptation overhead. Modern standards have largely migrated to probabilistic shaping (Ch. 19), which reaches similar gain with simpler architecture.