Chapter Summary

Chapter Summary

Key Points

• 1.

  A lattice $\Lambda \subset \mathbb{R}^n$ is a discrete additive subgroup of full rank, specified (up to unimodular basis change) by a generator matrix $\mathbf{G}$. The fundamental volume $V(\Lambda) = |\det \mathbf{G}|$ is basis-invariant; the Gram matrix $\mathbf{A} = \mathbf{G}^T \mathbf{G}$ satisfies $\det \mathbf{A} = V(\Lambda)^2$. The dual lattice $\Lambda^* = \{\mathbf{y} : \langle \mathbf{x}, \mathbf{y}\rangle \in \mathbb{Z}\}$ has generator matrix $\mathbf{G}^{-T}$ and volume $1/V(\Lambda)$.

• 2.

  Four geometric invariants do all the work: fundamental volume $V(\Lambda)$, packing radius $\rho = d_{\min}/2$, covering radius $R$, and kissing number $K$. From these we derive the packing density $\Delta(\Lambda) = \rho^n V_n / V(\Lambda)$, the center density $\delta(\Lambda) = \rho^n / V(\Lambda)$, and the Hermite ratio $\gamma(\Lambda) = d_{\min}^2 / V^{2/n}$.

• 3.

  The classical lattices — $\mathbb{Z}^n$, $A_n$, $D_n$, $E_6, E_7, E_8$, $K_{12}$, $\Lambda_{24}$ — are the benchmarks. Each is the densest known lattice in its dimension; $A_2$ is densest 2D (Gauss 1831), $D_4, D_5$ are densest in $n = 4, 5$ (Korkine–Zolotarev 1877), $E_8$ and $\Lambda_{24}$ are densest in $n = 8, 24$ (Viazovska 2017). Their kissing numbers grow from $2n$ (cube) to $196560$ (Leech).

• 4.

  The theta series $\Theta_\Lambda(q) = \sum_m N_m q^m$ packages the count of lattice points of each squared norm. The first nonzero coefficient after $N_0 = 1$ is the kissing number $K(\Lambda)$; higher coefficients feed the union-bound AWGN error analysis. For the integer lattice, $\Theta_{\mathbb{Z}^n}(q) = \theta_3(q)^n$; for $E_8$, $\Theta_{E_8} = E_4$ (the weight-4 Eisenstein series); for $\Lambda_{24}$, a specific weight-12 modular form.

• 5.

  Minkowski–Hlawka gives a lower bound $\Delta_n \ge \zeta(n) / 2^{n-1}$ by random-lattice averaging — the direct analogue of Shannon's random coding. Kabatiansky–Levenshtein gives the best-known dimension-independent upper bound $\log_2 \Delta_n \le -0.599 n + o(n)$. The gap is exponential in $n$ and is the central open problem of sphere-packing theory.

• 6.

  Viazovska ($2017$) proved $E_8$ and $\Lambda_{24}$ are globally optimal packings using a "magic" Schwartz function built from modular forms. These are the only $n > 3$ where the densest packing is exactly known. In all other dimensions, the best known lattice is conjecturally optimal but unproven; for coded-modulation design, use the best known lattice from the Nebe–Sloane database.

• 7.

  Every QAM constellation is a finite piece of $\mathbb{Z}^2$; every high-dimensional lattice code lives on a classical lattice. The fundamental coding gain $\gamma_c(\Lambda)$ over $\mathbb{Z}^n$ is what Ch. 4's coset-code framework uses; the Minkowski–Hlawka existence theorem is what Ch. 16's Erez–Zamir lattice code uses. Lattice theory is the absolute upper envelope for coded-modulation performance.