Chapter 15: Lattice Fundamentals

Learning Objectives

• State the formal definition of a lattice $\Lambda \subset \mathbb{R}^n$ as a discrete additive subgroup of full rank, and produce generator matrices $\mathbf{G}$ and dual-lattice bases $\mathbf{G}^{-T}$ for the canonical examples $\mathbb{Z}^n$, $A_n$, $D_n$, $E_8$, and $\Lambda_{24}$
• Compute the fundamental volume $V(\Lambda)$, packing radius $\rho(\Lambda)$, covering radius $R(\Lambda)$, kissing number $K(\Lambda)$, and center density $\delta(\Lambda)$ for low-dimensional lattices, and relate them through the identity $\Delta(\Lambda) = \rho^n V_n / V(\Lambda)$
• Write down the theta series $\Theta_\Lambda(q) = \sum_{\mathbf{x} \in \Lambda} q^{\|\mathbf{x}\|^2}$ of a lattice, compute its first few coefficients for $\mathbb{Z}^n$, $D_4$, and $E_8$, and use the Jacobi identity $\Theta_{\mathbb{Z}}(q) = \theta_3(q)$
• State the Minkowski-Hlawka lower bound $\Delta_n \ge \zeta(n) / 2^{n-1}$ on the densest lattice packing, and sketch the averaging-argument proof in the style of Shannon's random-coding argument
• Explain why the dimensions $n = 8$ (Viazovska 2017, $E_8$) and $n = 24$ (Cohn–Kumar–Miller–Radchenko–Viazovska 2017, $\Lambda_{24}$) are uniquely settled, while the densest packing in most other dimensions is still an open problem
• Translate lattice-theoretic quantities into coded-modulation gains: why QAM is a finite piece of $\mathbb{Z}^2$, why $A_2$ is the best 2D constellation by $0.6$ dB, and how coding gain $\gamma_c$ over $\mathbb{Z}^n$ propagates into Ch. 16–17 constructions