References & Further Reading

References

H. El Gamal, G. Caire, and M. O. Damen, Lattice Coding and Decoding Achieve the Optimal Diversity-Multiplexing Tradeoff of MIMO Channels, 2004
The foundational CommIT-group paper on LAST codes. Introduces the construction (nested lattice codes over MIMO) and proves DMT-optimality via MMSE-GDFE + lattice decoding. Section III is the main theorem; Section IV compares with CDA codes and V-BLAST.
K. R. Kumar and G. Caire, Space-Time Codes from Structured Lattices, 2009
The CommIT-group constructive companion to El Gamal-Caire-Damen (2004). Shows that replacing the random fine lattice by $E_8$, Leech, or Barnes-Wall preserves DMT-optimality and gains $3-6$ dB of finite-SNR coding gain. The conference version appeared in 2008; the journal version is 2009. Commonly cited as "Kumar-Caire 2008."
U. Erez and R. Zamir, Achieving $\tfrac{1}{2} \log(1 + \mathrm{SNR})$ on the AWGN channel with lattice encoding and decoding, 2004
The foundational proof that nested lattice codes with MMSE scaling and common random dithering achieve the AWGN capacity. Provides the "AWGN half" of the LAST argument — the MIMO side is El-Gamal-Caire-Damen's MMSE-GDFE triangularisation. Chapter 16 of this book covers the proof in detail.
L. Zheng and D. N. C. Tse, Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels, 2003
The canonical paper defining the diversity-multiplexing tradeoff for i.i.d. Rayleigh MIMO channels. Proves the Zheng-Tse curve $d^*(r) = (n_t - r)(n_r - r)$ for $r \\in \\{0, 1, \\ldots, \\min(n_t, n_r)\\}$, piecewise-linearly interpolated. The benchmark against which LAST codes are measured.
P. Elia, K. R. Kumar, S. A. Pawar, P. V. Kumar, H.-f. Lu, and G. Caire, Explicit Space-Time Codes Achieving the Diversity-Multiplexing Gain Tradeoff, 2006
The CommIT-group CDA-NVD paper (Ch. 13). Gives explicit DMT-optimal codes for arbitrary $(n_t, n_r)$ via cyclic division algebras with non-vanishing determinant. The algebraic counterpart to LAST's information-theoretic construction.
G. J. Foschini, Layered Space-Time Architecture for Wireless Communication in a Fading Environment When Using Multi-Element Antennas, 1996
Foschini's original diagonal-BLAST paper. The conceptual ancestor of V-BLAST and MMSE-SIC, which in turn inspired the MMSE-GDFE of LAST decoding.
P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel, 1998
The V-BLAST paper. Practical layered MIMO architecture with ZF or MMSE detection and successive interference cancellation. The precursor of the MMSE-GDFE receiver used in LAST decoding.
M. O. Damen, H. El Gamal, and G. Caire, On maximum-likelihood detection and the search for the closest lattice point, 2003
Companion paper to El Gamal-Caire-Damen 2004. Refines the sphere decoder complexity analysis ($O(M^{n_t T / 2})$ average at high SNR) and proposes the augmented-matrix/QR structure that became the MMSE-GDFE. Essential reference for sphere-decoder practitioners.
E. Viterbo and J. Boutros, A Universal Lattice Code Decoder for Fading Channels, 1999
The modern-form sphere decoder: a general closest-lattice-point algorithm applicable to any lattice. Essential for practical ML decoding of LAST and CDA codes in moderate dimensions. The LTE uplink receiver's decoder descends from this paper.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, 3rd ed., 1999
The encyclopaedic lattice-theoretic reference. Chapters 4-5 catalogue $E_8$, Leech, Barnes-Wall, and the root lattices that serve as inner lattices in structured LAST. Table 1.2 tabulates packing densities and kissing numbers.
R. Zamir, Lattice Coding for Signals and Networks, Cambridge University Press, 2014
Modern lattice-codes reference with networks emphasis. Chapter 9 covers the LAST construction in detail; Chapters 7-8 cover nested- lattice codes underlying LAST. The definitive pedagogical treatment of the material in this chapter.
J.-C. Belfiore, G. Rekaya, and E. Viterbo, The Golden Code: A $2 \times 2$ Full-Rate Space-Time Code with Non-vanishing Determinants, 2005
The Golden code paper. Introduces the $2 \\times 2$ full-rate DMT-optimal space-time code based on the golden-ratio algebraic extension. The CDA-NVD benchmark against which structured LAST codes are compared in §5.
D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005
Standard textbook. Chapter 8 covers MMSE-SIC and the MIMO sum capacity; Chapter 9 covers DMT and space-time codes. Written contemporaneously with the El Gamal-Caire-Damen 2004 paper and adopts many of its conventions.
B. Hassibi and B. M. Hochwald, High-rate codes that are linear in space and time, 2002
The Linear Dispersion Code (LDC) framework — the most general structure for linear space-time codes. CDA codes and LAST codes are both instances of this framework. Useful for placing LAST in the broader linear-STC landscape.
B. Hassibi and H. Vikalo, On the sphere-decoding algorithm I. Expected complexity, 2005
Detailed analysis of sphere-decoder expected complexity — polynomial at high SNR, exponential at low SNR. The theoretical basis for when sphere decoding is deployable and when it is not.
S. Tavildar and P. Viswanath, Approximately universal codes over slow-fading channels, 2006
Introduces the notion of approximate universality — a code that achieves the DMT under any fading distribution with regular density. Used in Ch. 13 to establish universality of CDA-NVD codes and in Ch. 17 discussions.
J. Boutros, G. Caire, E. Viterbo, and M. O. Damen, Good lattice constellations for both Rayleigh fading and Gaussian channels, 2003
Early CommIT-group work on lattice constellations robust to both fading and AWGN. Conceptual precursor to LAST — it explores the same design space in the single-antenna fading setting.
B. Nazer and M. Gastpar, Compute-and-forward: Harnessing interference through structured codes, 2011
The compute-and-forward paper. Extends LAST-style lattice codes to multi-user relay networks, with the relay decoding lattice equations instead of individual messages. Forward reference to Ch. 18 of this book.
D. S. Millar, T. Koike-Akino, M. Kojima, and K. Parsons, Experimental Demonstration of High Dimensional 4D-Set Partitioning, 2014
Experimental demonstration of $E_8$-coded coherent optical transmission at 200G — the practical deployment of lattice constellations in fiber-optic systems.
3GPP, NR; Physical channels and modulation (Release 16), 2020
5G NR physical layer specification. Section 6.3 defines the MIMO precoding codebook (Type I/II); Section 7.4 defines the reference signals. Confirms that 5G NR uses codebook-based precoding with CSIT, not LAST codes.