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Compute-and-Forward Strategies for Cooperative Distributed Antenna Systems

Song-Nam Hong, Giuseppe Caire

arXiv:1210.0160v2cs.IT

TL;DR

Distributed antenna systems must balance partial decoding or precoding against fixed-rate digital backhaul. This paper develops compute-and-forward-based uplink and downlink strategies, node-selection schemes, and integer-forcing beamforming, achieving competitive or sometimes better performance with lower complexity.

  • Problem

    Distributed antenna systems require partial decoding or precoding over digital backhaul links with fixed rate R0.

  • Method

    The paper combines CoF, Reverse CoF, low-complexity AT and UT selection, and Integer Forcing Beamforming for DAS uplink and downlink optimization.

  • Results

    The proposed strategies achieve similar and sometimes better performance than relevant comparison schemes, with significantly lower complexity.

  • Takeaways & Limitations

    Compute-and-forward-based DAS strategies provide competitive performance across uplink and downlink settings while addressing selection and backhaul constraints.

Abstract

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We study a distributed antenna system where $L$ antenna terminals (ATs) are connected to a Central Processor (CP) via digital error-free links of finite capacity $R_0$, and serve $K$ user terminals (UTs). We contribute to the subject in the following ways: 1) for the uplink, we apply the "Compute and Forward" (CoF) approach and examine the corresponding system optimization at finite SNR; 2) For the downlink, we propose a novel precoding scheme nicknamed "Reverse Compute and Forward" (RCoF); 3) In both cases, we present low-complexity versions of CoF and RCoF based on standard scalar quantization at the receivers, that lead to discrete-input discrete-output symmetric memoryless channel models for which near-optimal performance can be achieved by standard single-user linear coding; 4) For the case of large $R_0$, we propose a novel "Integer Forcing Beamforming" (IFB) scheme that generalizes the popular zero-forcing beamforming and achieves sum rate performance close to the optimal Gaussian Dirty-Paper Coding. The proposed uplink and downlink system optimization focuses specifically on the ATs and UTs selection problem. We present low-complexity ATs and UTs selection schemes and demonstrate, through Monte Carlo simulation in a realistic environment with fading and shadowing, that the proposed schemes essentially eliminate the problem of rank deficiency of the system matrix and greatly mitigate the non-integer penalty affecting CoF/RCoF at high SNR. Comparison with other state-of-the art information theoretic schemes, such as "Quantize reMap and Forward" for the uplink and "Compressed Dirty Paper Coding" for the downlink, show competitive performance of the proposed approaches with significantly lower complexity.

I. INTRODUCTION

The paper studies an intermediate distributed antenna architecture with partial AT decoding or precoding over fixed-rate digital backhaul, developing CoF/RCoF methods and low-complexity quantized implementations. It also addresses rank deficiency and non-integer penalties through system optimization and introduces IFB for large backhaul rates.

  • System model: The intermediate DAS architecture uses AT partial decoding or precoding and fixed-rate digital links R0, with analog forwarding unavailable.Quantization and forwarding are therefore required between ATs and the CP.
  • Core schemes: The paper applies CoF to the uplink at finite SNR and proposes the novel RCoF precoding scheme for the downlink.The uplink study includes general channels with fading and shadowing, beyond the Wyner model.
  • Low-complexity implementations: QCoF and RQCoF use standard scalar quantization to produce discrete-input discrete-output symmetric memoryless channels matched to standard single-user linear coding.They can be implemented with q-ary LDPC codes, with essentially linear complexity in code block length and polynomial complexity in the system size.
  • System optimization: The optimization targets CoF/RCoF non-integer penalties and finite-field system-matrix rank deficiency through power allocation, network decomposition, and AT or UT selection.In most practical cases, the AT and UT selection problems can be optimally solved by a simple greedy algorithm.
  • Large-backhaul downlink: For large R0, IFB achieves rates close to information-theoretic optimal Gaussian DPC and significantly outperforms conventional ZFB.IFB is introduced as a downlink scheme for the large-backhaul regime.

II. PRELIMINARIES · A. Distributed Antenna Systems: Channel Model · B. Nested Lattice Codes

The preliminaries define the distributed antenna system’s uplink and downlink channel models, power and channel assumptions, and the nested-lattice framework used for coding operations.

  • II. PRELIMINARIES: The preliminaries establish definitions and results used throughout the paper.
  • A. Distributed Antenna Systems: Channel Model: The system comprises L single-antenna ATs and K single-antenna UTs connected through rate-R0 digital backhaul links.
  • A. Distributed Antenna Systems: Channel Model: The uplink uses blocklength n with channel matrix H, additive i.i.d. Gaussian noise, and channel coefficients constant over the block and known to all nodes.
  • A. Distributed Antenna Systems: Channel Model: The downlink is modeled as ˜Y = ˜H ˜X + ˜Z, with AT codewords, UT outputs, Gaussian receiver noise, and downlink matrix ˜H.
  • A. Distributed Antenna Systems: Channel Model: Both uplink and downlink impose symmetric per-antenna power constraints because ATs and UTs are separately powered.
  • B. Nested Lattice Codes: Nested lattice coding uses Gaussian integers, a Gaussian prime p, finite-field operations over Fp2, and Construction A to form nested lattices.
  • B. Nested Lattice Codes: The construction yields Λ ⊆ Λ1 ⊆ p−1Λ with |Λ1/Λ| = p2r and |p−1Λ/Λ1| = p2(n−r).
  • B. Nested Lattice Codes: Lattice quantization and modulo operations define the Voronoi region and code L = Λ1 ∩ VΛ, whose rate is R = 1/n log p and whose messages map naturally to codewords.

C. Compute and Forward · III. COMPUTE AND FORWARD FOR THE DAS UPLINK

The paper recalls Compute-and-Forward (CoF), where lattice decoding recovers integer combinations at an achievable computation rate, and applies it to the DAS uplink. Each antenna terminal decodes and forwards a combination to the central processor, while network decomposition and antenna selection address rank and rate limitations.

  • C. Compute and Forward: CoF lattice decoding recovers an integer combination of user codewords at an achievable computation rate determined by the effective noise.The effective noise includes non-integer self-interference and additive Gaussian noise; minimizing its variance determines the computation rate.
  • C. Compute and Forward: The coefficient-optimization problem minimizes a positive-definite quadratic form over Gaussian-integer vectors, equivalent to finding a shortest lattice point.The shortest-point search can use complex LLL followed by Phost or Schnorr-Euchner enumeration.
  • III. COMPUTE AND FORWARD FOR THE DAS UPLINK: In the DAS uplink, CoF is applied independently at each antenna terminal, with each terminal selecting its target integer vector according to Algorithm 1.The exposition assumes K = L, while the notation also covers K < L when antenna selection is considered.
  • III. COMPUTE AND FORWARD FOR THE DAS UPLINK: Each antenna terminal can decode reliably when R ≤ R(hℓ, aℓ, SNR), then forwards its decoded information message to the central processor if R ≤ R0.The central processor collects the messages and forms linear equations over Fp2.
  • III. COMPUTE AND FORWARD FOR THE DAS UPLINK: Provided that Q has rank K over Fp2 and all terminals satisfy the computation-rate condition, the central processor recovers the user messages with arbitrarily small error probability.The resulting achievable per-user rate is given by the paper’s stated CoF expression.
  • III. COMPUTE AND FORWARD FOR THE DAS UPLINK: Decentralized coefficient selection can make Q rank-deficient, motivating selection of K antenna terminals whose coefficients form a full-rank system matrix.The selection problem is deferred to Section VII.
  • III. COMPUTE AND FORWARD FOR THE DAS UPLINK: Network decomposition permutes Q into diagonal blocks, allowing CoF to operate on independent subnetworks and summing their component sum rates instead of taking a global minimum.The decomposition also reduces Gaussian-elimination complexity by performing elimination independently for each subnetwork.
  • III. COMPUTE AND FORWARD FOR THE DAS UPLINK: Under the full-rank condition, Q can be permuted into diagonal blocks Q(As, Us), each of which is full-rank and square.Theorem 1 formalizes CoF with network decomposition for DAS uplinks whose system matrix has this block-diagonal structure.

IV. REVERSE COMPUTE AND FORWARD FOR THE DAS DOWNLINK

This section proposes Reverse Compute-and-Forward (RCoF), a downlink precoding scheme that eliminates finite-field interference at the Central Processor. Each user terminal then faces an individual computation-rate constraint determined by its own effective noise, subject to backhaul and system-matrix conditions.

  • RCoF principle: RCoF uses finite-field zero-forcing precoding at the Central Processor to eliminate the integer-valued interference that user terminals cannot share over backhaul links.The remaining effective noise comprises non-integer residual interference and Gaussian channel noise.
  • RCoF advantage: Unlike uplink CoF, each user terminal’s message is constrained by its own computation rate rather than the minimum computation rate across all receivers.This follows because each user terminal observes only its intended lattice codeword plus effective noise.
  • RCoF operation: RCoF assumes the downlink system matrix ˜Q has rank L and proceeds by precoding L messages before forwarding one precoded message to each antenna terminal.Each antenna terminal locally maps its received precoded message to a lattice codeword and transmits the corresponding channel input.
  • Achievable rates: Each intended message is reliably recoverable when Rℓ≤R(˜hkℓ, ˜akℓ, SNR), with scaling coefficients and integer vectors optimized independently at each user terminal.The precoded messages can be sent over the digital backhaul when R1≤R0.
  • Symmetric channels: When every channel-matrix row is a permutation of the first, all user terminals have the same computation rate, so one common lattice code suffices.A circulant channel matrix, as in the Wyner model, is given as an example.

V. LOW-COMPLEXITY SCHEMES

The section develops low-complexity CoF and RCoF schemes using one-dimensional lattices and receiver-side scalar quantization for finite-resolution ADCs. Component-wise modulo and quantization reduce the system to finite-field additive-noise channels suitable for coding and tractable coefficient optimization.

  • Low-complexity construction: One-dimensional lattices and scalar quantization enable low-complexity CoF and RCoF when receivers use fixed finite-resolution ADCs.Lattice quantization is unavailable because it requires unquantized soft samples, so scalar quantization is incorporated into the channel model.
  • Low-complexity construction: Component-wise modulo-Λ operations followed by scalar quantization yield a symbol-by-symbol channel model.The receiver applies Q(τ/p)Z[j](·) component-wise, exploiting the n-dimensional complex cubic lattice structure.
  • Equivalent discrete channels: The lattice encoders, G-MAC, and receiver mapping reduce to an equivalent discrete linear additive-noise finite-field MAC.The induced channel is the FF-MAC in (25), with discrete additive noise determined by the channel, integer coefficient vector, and scaling parameter.
  • Equivalent discrete channels: Modulo and scalar quantization commute, allowing analog wrapping into the Voronoi region of τZ[j] before quantization.This implementation is an analog sawtooth transformation followed by scalar quantization of the real and imaginary signal components.
  • Parameter optimization: Receiver-specific integer search selects the coefficient vector and scaling parameter by minimizing the discrete additive-noise entropy surrogate.Direct entropy minimization is not tractable, so the schemes use the integer search of Algorithm 1 independently at each receiver.

A. QCoF and LQF for the DAS Uplink · B. RQCoF for the DAS Downlink

The uplink section develops QCoF and LQF as quantized compute-and-forward schemes with different AT processing, while the downlink section introduces RQCoF using finite-field matrix-inversion precoding. Both sections characterize achievable rates or capacity through discrete finite-field channels and linear coding.

  • A. QCoF and LQF for the DAS Uplink: QCoF and LQF provide two uplink schemes distinguished by how antenna terminals process received signals.QCoF is described as a low-complexity quantized version, while LQF uses a different AT processing approach.
  • A. QCoF and LQF for the DAS Uplink: QCoF converts each antenna-terminal observation into a discrete additive-noise channel over Fp², enabling decoding when R ≤ 2 log p − H(ζ(hℓ, aℓ)).Decoded finite-field combinations are forwarded to the central processor, which recovers the original user messages by Gaussian elimination.
  • A. QCoF and LQF for the DAS Uplink: QCoF with network decomposition achieves a sum-rate expression proportional to |As| and limited by min{R0, min{2 log p − H(ζ(hk, ak))}}.The theorem applies QCoF to a DAS uplink with channel matrix H.
  • A. QCoF and LQF for the DAS Uplink: LQF forwards quantized channel observations directly to the central processor, forming a finite-field multiple-access channel without local decoding or binning.Under 2 log p ≤ R0, its achievable sum rate can be attained by linear coding.
  • A. QCoF and LQF for the DAS Uplink: The relative performance of QCoF and LQF depends on R0, p, and the channel realization H.QCoF is favored by sufficiently large p in symmetric channels, whereas LQF outperforms QCoF when p is predetermined and relatively small compared with R0.
  • B. RQCoF for the DAS Downlink: RQCoF applies finite-field matrix-inversion precoding at the central processor for the DAS downlink.The resulting finite-field broadcast channel has capacity achievable by simple matrix inversion when the precoding matrix has full rank.
  • B. RQCoF for the DAS Downlink: Theorem 6 summarizes RQCoF for a DAS downlink with channel matrix ˜H = [˜h1, . . . , ˜hL]T ∈ C^L×L.The scheme is presented for the downlink channel after finite-field matrix-inversion precoding.
  • B. RQCoF for the DAS Downlink: For each downlink message, the intended user terminal can recover its codeword when Rℓ ≤ 2 log p − H(ζ(˜hkℓ, ˜akℓ)).RQCoF uses nested linear codes and scalar quantization at the user terminals after central-processor precoding.

VI. COMPARISON WITH KNOWN SCHEMES ON THE WYNER MODEL · A. Review of some Classical Coding Strategies · 1) QMF:

The paper compares DAS coding strategies on the analytically tractable symmetric Wyner model, including QMF and DF for the uplink and CDPC and CZFB for the downlink. QMF uses quantization and binning at the ATs, approaches the underlying multi-antenna G-MAC sum rate as R0 grows, and reaches LR0 at high SNR with fixed R0.

  • VI. COMPARISON WITH KNOWN SCHEMES ON THE WYNER MODEL: The comparison uses the symmetric Wyner model because of its simplicity and analytic tractability.
  • VI. COMPARISON WITH KNOWN SCHEMES ON THE WYNER MODEL: The uplink comparison includes Quantize reMap and Forward (QMF) and Decode and Forward (DF), while the downlink comparison includes CDPC and CZFB.
  • A. Review of some Classical Coding Strategies: QMF vector-quantizes each AT’s received signal at rate R′ ≥ R0, hashes the quantization bits into length-nR0 words, and jointly decodes the UT messages at the CP.
  • 1) QMF:: As R0 →∞, RQMF tends to the sum rate of the underlying multi-antenna G-MAC channel with L users and one L-antennas receiver.
  • 1) QMF:: At SNR →∞ with fixed R0, RQMF → LR0.
  • 1) QMF:: Without binning, QF forwards the quantization bits directly to the CP using quantization rate R′ = R0.
  • 1) QMF:: In the Wyner model, each AT sees a three-input G-MAC from UTs ℓ−1, ℓ, and ℓ+1, yielding Rsum = L × min{max(R1, R2), R0}.The local decoding alternatives are treating interference as noise or decoding all messages at each AT.
  • 1) QMF:: QMF has no joint-processing gain, but can be optimal when R0 is sufficiently small relative to wireless-channel rates or when γ is very small.DF is implemented in networks of small cells, motivating comparison with current technology.

3) CDPC: · 4) CZFB: · B. Numerical Results

CDPC and CZFB provide compressed downlink baselines whose quantization noise and finite-backhaul effects shape their rates, while numerical results show CoF/RCoF advantages at limited backhaul and high SNR. Power allocation reduces CoF’s integer penalty, but its extension beyond structured channels remains unclear.

  • 3) CDPC:: CDPC compresses jointly DPC-precoded codewords for the ATs, which also transmit quantization noise, and is generally suboptimal at finite R0.It is expected to be near optimal only for large R0.
  • 3) CDPC:: CDPC’s achievable sum rate equals the sum capacity of the modified vector BC with per-antenna power constraints and quantization-noise variance.The rate can be computed using an efficient Lagrangian-duality algorithm.
  • 4) CZFB:: CZFB uses the inverse channel matrix B = ˜H−1 to precoding-compress ZFB signals, while AT quantization noise again contributes to effective receiver noise.Its useful signal power is SNR 2R0−1 2R0∥bℓ∥2 under the stated non-unitary precoding.
  • B. Numerical Results: Power allocation assigns odd terminals βP and even terminals (2−β)P, reverses their roles across slots, and optimizes β for integer approximation.This preserves each terminal’s average power constraint while creating more favorable effective coefficients.
  • B. Numerical Results: For SNR = 25 dB and L = ∞, power allocation significantly reduces the integer penalty and almost achieves the cut-set bound for R0 ≤7 bits.QCoF with p = 251 approaches the corresponding high-dimensional scheme within ≈0.5 bit per complex dimension.
  • B. Numerical Results: RCoF outperforms CDPC for R0 ≤6.5 bits per channel use, while quantized QCoF and RQCoF outperform DF and CZFB, respectively.The downlink RCoF sum rate uses separate lattice codes for odd- and even-numbered users.
  • B. Numerical Results: CoF and RCoF are strong DAS uplink and downlink candidates, especially at small-to-moderate R0 and high SNR, where limited backhaul becomes the bottleneck.Their gains are attributed partly to mitigating the non-integer penalty.
  • B. Numerical Results: Power allocation exploits symmetric channels, but its extension to fading, shadowing, and pathloss channels is unclear; AT/UT selection is proposed to improve general-network performance.The next section addresses random channel coefficients and multiuser diversity.

VII. ANTENNA AND USER SELECTION · A. Antenna Selection for the DAS Uplink · 1) AT selection for CoF (or QCoF):

The section formulates antenna and user selection for DAS operation with equal numbers of active ATs and UTs, then develops uplink AT-selection methods for CoF/QCoF. The proposed greedy procedure is optimal for dense, non-decomposable networks and supports efficient processing of decomposed subnetworks.

  • VII. ANTENNA AND USER SELECTION: Each scheduling slot selects active UT and AT subsets, with the proposed schemes requiring equal numbers of active terminals.Active UTs transmit or receive during the current slot, while active ATs receive or transmit, respectively.
  • A. Antenna Selection for the DAS Uplink: For the DAS uplink, the active UT set is fixed, U = [1 : K], with K < L, and the system selects K active ATs.The selected AT subset A has cardinality K.
  • A. Antenna Selection for the DAS Uplink: Each AT independently chooses integer coefficients and q_ℓ using Algorithm 1 to maximize its computation rate R_ℓ = R(h_ℓ, a_ℓ, SNR).The CP knows the resulting {q_ℓ, R_ℓ} for all ATs.
  • A. Antenna Selection for the DAS Uplink: The CP selects ATs to maximize sum rate while ensuring that the resulting system matrix is full-rank for the given active UT set.This selection uses the AT-provided coefficient vectors and computation rates.
  • 1) AT selection for CoF (or QCoF):: In general, the CoF/QCoF AT-selection optimum requires exhaustive search over all |U|×|U| submatrices of Q([1 : L], U).The problem has no particularly nice structure in general.
  • 1) AT selection for CoF (or QCoF):: For dense networks whose Q([1 : L], U) cannot be decomposed into block-diagonal form, a low-complexity greedy algorithm finds an optimal AT selection.If an optimal solution has S(A⋆) = 1, Algorithm 2 finds it.
  • 1) AT selection for CoF (or QCoF):: When multiple disjoint subnetworks exist, the method decomposes the network, applies Algorithm 2 to each component, and is generally suboptimal but efficient.If each component’s optimum requires no further decomposition, Lemma 3 guarantees an optimal global solution.
  • 1) AT selection for CoF (or QCoF):: Ignoring network decomposition, Algorithm 2 uses m = L, n = K, Q = Q([1 : L], U), and weights w_ℓ = min{R0, R_ℓ} to find the maximum computation rate.Lemma 3 states that Algorithm 2 finds a solution when Rank(Q) = n.

2) AT selection for LQF: · B. User Selection for the DAS Downlink · C. Comparison on the Bernoulli-Gaussian Model

The paper formulates AT and UT selection for LQF and DAS downlink schemes as linear optimization over matroid constraints, solved optimally by greedy selection. In Bernoulli-Gaussian simulations, greedy selection improves rates, mitigates rank deficiency and non-integer penalties, and preserves competitive performance with lower-complexity implementations.

  • 2) AT selection for LQF:: LQF AT selection maximizes a linear objective over a matroid whose independent sets produce linearly independent system-matrix rows.The weights are w_ℓ = min{R0, R_ℓ}.
  • 2) AT selection for LQF:: The matroid greedy algorithm is optimal for LQF AT selection and coincides with Algorithm 2 using Q = Q([1 : L], U).The selection weights are w_ℓ = min{R0, R_ℓ}.
  • B. User Selection for the DAS Downlink: With all L ATs fixed and K > L, downlink user selection chooses L users whose system matrix has rank L while maximizing the DAS downlink sum rate.The objective uses individual RCoF or RQCoF computation rates.
  • B. User Selection for the DAS Downlink: Downlink UT selection is likewise a linear maximization over a matroid, so Algorithm 2 with wk = min{R0, ˜Rk} gives an optimal solution.The method applies to both RCoF and RQCoF rate objectives.
  • C. Comparison on the Bernoulli-Gaussian Model: The Bernoulli-Gaussian model combines i.i.d. Rayleigh fading with Bernoulli path blocking, using P(γℓ,k = 1) = q, and treats rank deficiency as an information-outage event.Ergodic sum rates are obtained by Monte Carlo averaging; a rank-deficient realization has zero instantaneous sum rate.
  • C. Comparison on the Bernoulli-Gaussian Model: For the uplink, greedy AT selection substantially outperforms random selection and essentially eliminates rank deficiency when L ≫ K.Selecting 5 ATs out of 25 can outperform using all 25 ATs in the reported example.
  • C. Comparison on the Bernoulli-Gaussian Model: For the downlink, greedy UT selection achieves a clear multiuser-diversity gain over random selection while both methods exhibit the same sum-rate-versus-SNR slope.Greedy selection also chooses channel vectors whose coefficients are well approximated by integers for RCoF.

VIII. INTEGER-FORCING BEAMFORMING FOR THE HIGH-CAPACITY BACKHAUL CASE

In the infinite-backhaul downlink, Integer-Forcing Beamforming (IFB) chooses precoding to create an integer-valued effective channel, eliminating RCoF’s non-integer penalty while introducing a power penalty. By optimizing the integer matrix, IFB improves over zero-forcing beamforming and approaches dirty-paper coding sum capacity.

  • IFB construction: IFB chooses B = H̃^-1à so the effective channel H̃B equals the integer matrix Ã, enabling RCoF without a non-integer penalty.The integer matrix à may be general rather than the identity, distinguishing IFB from ZFB.
  • IFB construction: IFB removes RCoF’s non-integer penalty but incurs a power penalty from its non-unitary precoding matrix B.Restricting à = I makes IFB coincide with ZFB.
  • Optimization: Allowing a general integer matrix à makes IFB at least as good as ZFB and usually significantly better by reducing the power penalty.The optimization of à is difficult, so the paper uses sum-power minimization and lattice-reduction methods including complex LLL.
  • Decoding: Each user decodes its own lattice code without multiuser interference because RCoF eliminates the resulting integer-valued interferences.The effective channel is a G-MAC with integer channel coefficients.
  • Performance: ≈0.5 bits per user: IFB approaches DPC’s sum capacity within this gap while significantly improving over ZFB in the 5-antenna, 5-user Rayleigh-fading setting.The comparison uses infinite backhaul capacity and i.i.d. CN(0, 1) channel entries.

IX. CONCLUSIONS

The paper develops CoF-based uplink and RCoF-based downlink strategies with AT/UT selection, ADC-aware lattice coding, and IFB for large R0. These methods offer competitive performance and practical centralized processing when finite-rate backhaul limits system sum rate.

  • DAS uplink: For the uplink, CoF uses network decomposition and greedy AT selection for a given set of active UTs.
  • DAS downlink: For the downlink, RCoF reverses the AT/UT roles and uses finite-field linear precoding to eliminate multiuser interference.
  • DAS downlink: Downlink optimization selects UT subsets for active ATs; the resulting linear maximization under a matroid constraint is solved optimally by a greedy procedure.
  • Low-complexity and large-R0 regimes: ADC-aware lattice strategies enable simple single-user linear coding over Fq with q = p^2, while IFB generalizes zero-forcing beamforming without further non-integer penalty.
  • Performance and practicality: The proposed strategies achieve similar or sometimes better performance than QMF and CDPC in relevant regimes, with a clearer path to practical implementation.They can outperform decode-and-forward uplink and compressed linear beamforming downlink approaches at similar or better complexity.

APPENDIX A GAUSSIAN APPROXIMATION · APPENDIX B · APPENDIX C

The appendices derive a symmetric, finitely computable Gaussian-induced noise pmf, establish the finite-field MAC sum-rate result, and prove the corresponding decoupled finite-field BC sum-capacity characterization.

  • APPENDIX A GAUSSIAN APPROXIMATION: Gaussian symmetry gives equal probabilities for nonzero noise values paired by g(β1) + g(β2) = p.The pmf is obtained from the symmetry of the Gaussian distribution about the origin.
  • APPENDIX A GAUSSIAN APPROXIMATION: The Gaussian-induced noise pmf can be computed using Φ(x), whose rapid decay makes the required summations finite.The appendix reports that summing over m = 0, 1, 2 suffices for all numerical results.
  • APPENDIX B: For the FF-MAC y = Qx ⊕ ζ, the capacity region is formed by rate constraints indexed by every user subset S.The channel is defined over finite-field inputs x = (x1, ..., xK)^T, and the constraints are given in (67).
  • APPENDIX B: Because each fixed-input-distribution region is a polymatroid, its maximum sum rate lies on the dominant face.The appendix uses this structure to optimize the FF-MAC sum rate.
  • APPENDIX B: Time-sharing is unnecessary for maximum sum rate, and full-rank Q with uniform i.i.d. inputs achieves the sum rate in (29).The conclusion follows from the Markov relation q → x → y and the entropy inequality involving the additive noise components.
  • APPENDIX C: For the FF-BC y = ˜Qx ⊕ ζ, decoupling makes the sum rate equal to the sum of the individual channel capacities.This remains true regardless of statistical dependence across the noise components.
  • APPENDIX C: Uniformly distributed independent finite-field inputs achieve each decoupled channel capacity, so the sum rate in (31) is achievable and equals the FF-BC sum capacity.The converse uses single-user upper bounds, additive-noise mutual information, and H(yℓ) ≤ 2 log p.
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