Chapter 4: Coset Codes and Lattice-Based Coded Modulation

Learning Objectives

• State the definition of a lattice $\Lambda \subset \mathbb{R}^n$ via a generator matrix, and compute its fundamental volume $V(\Lambda)$, packing radius, covering radius, and kissing number for the canonical examples $\mathbb{Z}^n, D_n, E_8, \Lambda_{24}$
• Construct Forney's coset code $C(\Lambda/\Lambda'; \mathcal{C})$ as a binary code selecting cosets in a lattice partition, and compute its (fundamental) coding gain $\gamma_c$ from the partition-chain parameters
• Explain the decomposition of the gap to capacity into an (additive-dB) coding gain $\gamma_c$ and shaping gain $\gamma_s$, and prove the ultimate shaping-gain bound $\gamma_s \le \pi e / 6 \approx 1.53$ dB
• Construct a Voronoi constellation from a coding lattice $\Lambda_c$ and a shaping lattice $\Lambda_s$ and relate its shaping gain to the normalised second moment $G(\Lambda_s)$
• Describe two practical shaping schemes — shell mapping (Laroia–Farvardin–Tretter) and trellis shaping (Forney) — and quantify the rate-overhead / gain tradeoff each one pays
• Place coset codes in the chronological arc from Ungerboeck TCM (Ch. 2) to LAST codes (Ch. 17) and probabilistic shaping (Ch. 19)