References & Further Reading

References

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, 3rd ed., 1999
The encyclopaedic reference for lattice theory — the "bible" of the subject. Chapter 1 introduces lattices and their invariants; Chapter 2 covers Voronoi regions, theta series, and the second moment; Chapters 4–6 catalogue the root lattices, the Leech lattice, and the Niemeier classification; Chapter 20 gives practical encoding/decoding algorithms. Indispensable for any serious study of lattice codes.
M. S. Viazovska, The sphere packing problem in dimension 8, 2017
The breakthrough 22-page paper proving that $E_8$ achieves the densest sphere packing in $\\mathbb{R}^8$. Uses modular forms of half-integer weight to construct the Cohn–Elkies magic function. Viazovska was awarded the Fields Medal in 2022 for this work.
H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, The sphere packing problem in dimension 24, 2017
Companion paper to Viazovska (2017). Adapts the modular-form magic-function technique to dimension 24, proving that $\\Lambda_{24}$ (the Leech lattice) achieves the densest sphere packing in $\\mathbb{R}^{24}$.
H. Cohn and N. Elkies, New upper bounds on sphere packings I, 2003
Introduces the linear-programming method for sphere-packing upper bounds via Poisson summation. The method Viazovska sharpened to equality in dimensions 8 and 24.
G. A. Kabatiansky and V. I. Levenshtein, Bounds for packings on a sphere and in space, 1978
The asymptotic upper bound $\\log_2 \\Delta_n \\le -0.599 n + o(n)$ via LP bounds on kissing configurations. The best-known dimension-independent upper bound until Cohn–Elkies refinement and Viazovska closure in $n = 8, 24$.
J. Leech, Notes on sphere packings, 1967
Leech's original construction of the 24-dimensional lattice $\\Lambda_{24}$, based on the extended binary Golay code $\\mathcal{C}_{24}$. Proved optimal 50 years later by Cohn–Kumar– Miller–Radchenko–Viazovska (2017).
G. D. Forney Jr., Coset codes — part I: introduction and geometrical classification, 1988
Foundational paper of the lattice-coset-code framework. Introduces the partition-chain view and the fundamental coding gain for lattice-based TCM. Heavily referenced throughout this chapter.
G. D. Forney Jr., Coset codes — part II: binary lattices and related codes, 1988
Companion to Part I. Catalogs the binary lattices $\\mathbb{Z}^n$, $D_n$, $E_8$, and the Leech lattice $\\Lambda_{24}$ and the binary partitions between them.
G. D. Forney Jr. and G. Ungerboeck, Modulation and coding for linear Gaussian channels, 1998
Sweeping survey of coded modulation and lattice codes. Section V covers theta-series error analysis; Section II covers the fundamental coding-gain computation. Written by two of the principal architects of modern coded modulation.
E. Viterbo and J. Boutros, A universal lattice code decoder for fading channels, 1999
The sphere decoder: a general closest-lattice-point algorithm for arbitrary lattices. Essential for practical ML decoding of lattice codes in moderate dimensions.
R. Zamir, Lattice Coding for Signals and Networks, Cambridge University Press, 2014
Modern lattice-codes reference with a networks emphasis. Chapter 2 covers the foundations of lattice geometry used in this chapter; Chapters 7–9 cover nested-lattice schemes underlying LAST codes (Ch. 17 of the present book).
H.-A. Loeliger, Averaging bounds for lattices and linear codes, 1997
Connects the Minkowski–Hlawka random-lattice averaging to Shannon's random-coding arguments for the AWGN channel. The definitive reference for the informational analogy drawn in s05.
U. Erez and R. Zamir, Achieving $\tfrac{1}{2} \log(1 + \mathrm{SNR})$ on the AWGN channel with lattice encoding and decoding, 2004
First proof that lattice codes (with MMSE scaling and dithering) achieve the AWGN capacity. Forward reference for Ch. 16 of this book.
H. El Gamal, G. Caire, and M. O. Damen, Lattice coding and decoding achieve the optimal diversity–multiplexing tradeoff of MIMO channels, 2004
The CommIT group's foundational paper on Lattice Space-Time (LAST) codes. Shows that nested-lattice constructions achieve the optimal DMT of MIMO fading channels. Forward reference for Ch. 17.
T. C. Hales, A proof of the Kepler conjecture, 2005
Hales's proof that the face-centred cubic $D_3$ is the densest sphere packing in $\\mathbb{R}^3$. First of the global sphere-packing optimality proofs; Viazovska's 8D and 24D successors came 12 years later. A computer-assisted proof, formally verified in 2014.
E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of Abelian groups, 1959
Introduces the Barnes–Wall lattices $\\Lambda_{2^k}$, notable examples of dense lattices in $n = 16, 32, 64$ constructed from Reed–Muller codes. Used as default coding lattices in pre-Leech high-dimensional applications.
T. M. Cover and J. A. Thomas, Elements of Information Theory, Wiley-Interscience, 2nd ed., 2006
Standard reference for the Gaussian max-entropy theorem (Thm. 8.6.5) and differential-entropy tools used implicitly in s05's random-lattice argument.
C. F. Gauss, Recension der 'Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen' von Ludwig August Seeber, 1831
Gauss's review of Seeber's book on ternary quadratic forms. Contains the first proof that the hexagonal lattice $A_2$ is the densest 2D lattice packing — the starting point of lattice- density theory.
C. G. J. Jacobi, Fundamenta nova theoriae functionum ellipticarum, Königsberg: Borntraeger, 1829
Jacobi's treatise introducing the theta functions $\\theta_1, \\theta_2, \\theta_3, \\theta_4$ and the formulas for sums of squares. The direct antecedent of the modular-form theory Viazovska would use 188 years later.