References & Further Reading

References

B. Nazer and M. Gastpar, Compute-and-forward: Harnessing interference through structured codes, 2011
THE central paper of this chapter. Introduces the compute-and-forward framework: relay decodes lattice linear combinations $\sum_k a_k \mathbf{u}_k \bmod \Lambda_s$ at the Nazer-Gastpar rate $R(\mathbf{h}, \mathbf{a})$. Unifies physical-layer network coding, interference-channel inner bounds, and distributed source coding under a single lattice-coding framework. Proves the central rate formula (Thm. 1) and applies it to the two-way relay, the symmetric Gaussian interference channel, and Gaussian network coding.
S. Zhang, S. C. Liew, and P. P. Lam, Hot topic: physical-layer network coding, 2006
The original physical-layer network coding paper. Four-page MobiCom "hot topic" that proposed the analog-XOR two-slot TWRC protocol. Launched PNC as a subfield; over 2000 citations. Foundational reference for §4; equivalent to the Nazer-Gastpar $\mathbf{a} = (1, 1)^T$ special case (Thm. <a href="#thm-pnc-as-cf-special-case" class="ferkans-ref" title="Theorem: PNC Is the $\mathbf{a} = (1, 1)^T$ Special Case of CF" data-ref-type="theorem">TPNC Is the $\mathbf{a} = (1, 1)^T$ Special Case of CF</a>).
W. Nam, S.-Y. Chung, and Y. H. Lee, Capacity of the Gaussian two-way relay channel to within 1/2 bit, 2010
Proves that compute-and-forward with $\mathbf{a} = (1, 1)^T$ achieves the capacity region of the symmetric Gaussian two-way relay channel within $\tfrac{1}{2}$ bit per user, for all SNR. Foundational reference for §3; the constant-gap-to-capacity result for TWRC.
O. Ordentlich, U. Erez, and B. Nazer, The approximate sum capacity of the symmetric Gaussian $K$-user interference channel, 2014
Extends CF to the $K$-user Gaussian interference channel and introduces the integer-forcing linear receiver for MIMO. Proves the constant-gap-to-sum-capacity result (Thm. <a href="#thm-integer-forcing" class="ferkans-ref" title="Theorem: Integer-Forcing Achieves a Constant Gap to MIMO Sum-Capacity" data-ref-type="theorem">TInteger-Forcing Achieves a Constant Gap to MIMO Sum-Capacity</a>) and establishes CF as a general lattice-coding primitive beyond the TWRC. Awarded IEEE Information Theory Society Best Paper Award 2016. Foundational reference for §5.
U. Erez and R. Zamir, Achieving 1/2 log(1 + SNR) on the AWGN channel with lattice encoding and decoding, 2004
The foundational nested-lattice AWGN capacity result (Ch. 16 prerequisite). Proves that nested lattice codes with MMSE scaling achieve the single-user Gaussian capacity $\tfrac{1}{2} \log_2(1 + \ntn{snr})$. The Nazer-Gastpar theorem generalises this to multiple transmitters and arbitrary integer coefficients; the proof pattern (MMSE scaling, mod-$\Lambda_s$ reduction, Poltyrev-good $\Lambda_c$) carries over directly.
R. Zamir, Lattice Coding for Signals and Networks, Cambridge University Press, 2014
The definitive textbook on lattice coding for Gaussian channels and networks. Chapter 12 covers compute-and-forward in depth; Chapter 13 covers distributed source coding. Pedagogically complementary to this chapter, with more space for proofs and examples. Ties §1-§5 together.
B. Nazer and M. Gastpar, Computation over multiple-access channels, 2007
Nazer and Gastpar's original paper on computing functions over a MAC — the conceptual precursor to the 2011 compute-and-forward framework. Focuses on general functions (not just linear), uses random-coding arguments rather than lattices, and establishes the "function computation" viewpoint that the 2011 paper made concrete for linear functions.
R. Zamir, S. Shamai, and U. Erez, Nested linear/lattice codes for structured multiterminal binning, 2002
The nested-lattice framework for multiterminal problems. Established the $\Lambda_c \supset \Lambda_s$ construction with quantisation and dithering that underlies the Nazer-Gastpar achievability. Key precursor for §2's proof steps on shared lattice codebooks across users.
B. Nazer and M. Gastpar, Lattice coding increases multicast rates for Gaussian multiple-access networks, 2008
Conference-paper precursor to the 2011 IT paper. Showed that lattice coding outperforms random coding for the $K$-user multicast problem, motivating the full compute-and-forward framework.
M. P. Wilson, K. Narayanan, H. D. Pfister, and A. Sprintson, Joint physical layer coding and network coding for bidirectional relaying, 2010
Independent, contemporaneous derivation of the TWRC constant-gap result. Uses lattice codes with integer coefficients and proves within-constant-gap achievability for asymmetric TWRC. Companion to Nam-Chung-Lee 2010.
P. Popovski and H. Yomo, The anti-packets can increase the achievable throughput of a wireless multi-hop network, 2006
Independent, contemporaneous proposal of analog network coding for wireless two-way relay — published the same year as Zhang-Liew-Lam at a different venue. The "anti- packet" framing emphasises the throughput-gain argument: each user's transmission effectively conveys **two** packets (its own and its negative contribution to the other's).
T. J. Oechtering and M. Skoglund, Coding for the Gaussian two-way relay channel, 2009
Early information-theoretic analysis of the Gaussian TWRC achievability region. Predates the Nam-Chung-Lee $\tfrac{1} {2}$-bit theorem but establishes the rate-region structure and the MA-BC slot decomposition that §3 uses.
J. Zhan, B. Nazer, U. Erez, and M. Gastpar, Integer-forcing linear receivers, 2014
Comprehensive treatment of the integer-forcing receiver for MIMO, including the optimal-integer-matrix search via LLL reduction and bounds on the constant-gap performance. Extended companion to Ordentlich-Erez-Nazer 2014 with full proofs and engineering analysis. Source for §5's algorithm block.
J. Körner and K. Marton, How to encode the modulo-two sum of binary sources, 1979
The classical distributed-source-coding problem whose CF-lattice reformulation is discussed in §5. Körner and Marton showed that the sum-rate for jointly encoding the XOR of two correlated binary sources is $H(U_A \oplus U_B)$, not $H(U_A) + H(U_B)$. The lattice-CF counterpart achieves the analogous result on Gaussian sources.
R. F. H. Fischer and C. Siegl, Reconstruction and decoding for constellation-constrained compute-and-forward, 2014
Practical CF implementation with finite-alphabet constellations (4-QAM, 16-QAM) and LDPC outer codes. Discusses the integer-coefficient search complexity at moderate $K$ and the PNC-to-lattice-CF transition. Reference for §5's practical-deployment discussion.
K. Koike-Akino, P. Popovski, and V. Tarokh, Optimized constellations for two-way wireless relaying with physical network coding, 2009
Practical PNC with higher-order constellations. Shows how 16-QAM and 64-QAM PNC require Voronoi-optimised constellations to avoid the asymmetric-channel failure mode. Cited in §4's discussion of PNC limitations.
M. Ajtai, Generating hard instances of lattice problems, 1996
The NP-hardness result for SVP under randomised reductions. Cited in §5 to explain why the integer-coefficient search is computationally hard in general. Foundational lattice- cryptanalysis reference.
A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, 1982
The LLL lattice reduction algorithm. Polynomial-time approximation to the shortest-vector problem within a $2^{K/2}$ factor. The workhorse for integer-coefficient search in §5.
D. Wübben and P. Rost, Compute-and-forward relaying in cooperative satellite communications, 2015
CF prototype in a two-way satellite relay. Demonstrates a $\sim 2\times$ throughput gain at $\ntn{snr} = 15$ dB over decode-and-forward. Cited in §2 and §5 for the practical-deployment discussion.
T. M. Cover and A. El Gamal, Capacity theorems for the relay channel, 1979
The classical paper on the relay channel; introduces decode-and-forward, amplify-and-forward, and compress-and- forward as canonical strategies. CF is contrasted against these in §1 and §3. The cut-set upper bound used in Thm. <a href="#thm-nam-chung-suh-half-bit" class="ferkans-ref" title="Theorem: Nam-Chung-Lee: CF Is Within 1/2 Bit of TWRC Capacity" data-ref-type="theorem">TNam-Chung-Lee: CF Is Within 1/2 Bit of TWRC Capacity</a> is the Cover-El Gamal max-flow min-cut argument.
C. E. Shannon, Two-way communication channels, 1961
Shannon's original two-way channel paper — the foundational network-information-theory reference for §3's TWRC. Shannon's inner bound on the general two-way channel capacity is still not known to be tight; the relay-mediated version of §3 has a cleaner answer.
R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, Network information flow, 2000
The foundational network-coding paper. Proved that linear coding at intermediate nodes achieves the multicast capacity of wired networks — the digital precursor to the analog PNC and CF of this chapter. Historical context in §1 and §4.
3GPP, NR; Overall description; Stage 2, 2022. [Link]
5G NR overall architecture specification, including Integrated Access and Backhaul (IAB, Release 16). Cited in §1 and §3 to note that 5G NR does **not** implement compute-and-forward — IAB uses orthogonal multiplexing instead, though CF would structurally fit.
V. R. Cadambe and S. A. Jafar, Interference alignment and degrees of freedom of the $K$-user interference channel, 2008
Interference alignment: the $K$-user GIC achieves $K/2$ degrees of freedom via continuous-channel subspace alignment. CF gives an alternative achievability path for the GIC (Ordentlich-Erez-Nazer 2014) that does not require infinite channel diversity. Cited in §5 as the contrasting paradigm.
R. H. Etkin, D. N. C. Tse, and H. Wang, Gaussian interference channel capacity to within one bit, 2008
The 1-bit gap capacity result for the 2-user Gaussian interference channel. Benchmark against which the Ordentlich-Erez-Nazer 2014 CF-based bound is compared in §5. Established the constant-gap-to-capacity style of analysis that the Nazer-Gastpar framework extends.