Ring Compute-and-Forward Over Block-Fading Channels

Lyu, Shanxiang; Campello, Antonio; Ling, Cong

doi:10.1109/tit.2019.2927453

Cited by 14 publications

(7 citation statements)

References 51 publications

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“…As a comparison, our proof generalizes the analysis of [33] from Gaussian integers to general rings in quadratic fields. The proof also resonates with our Diaphantine approximation analysis in [31]. The difference is that the algebraic lattices here naturally reside in a C n space, while [31] employs embeddings to rise the lattices to a R n space.…”

Section: Appendix a Rotations And Quaternionssupporting

confidence: 63%

“…Although our work implies the infeasibility of addressing lattice reduction for non-Euclidean imaginary quadratic domains directly, lattices over such domains can always be transformed to real lattices and subsequently associating with real reduction algorithms. An interesting future direction is to study the reduction theory on more complicated Humber-form lattices in [31], which arises from C&F over block-fading channels. Another area to explore is reduction over "trace-Euclidean" domains, i.e.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Humber-form lattices in [31], which arises from C&F over block-fading channels. Another area to explore is reduction over "trace-Euclidean" domains, i.e.…”

Section: Discussionmentioning

confidence: 99%

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Performance Limits of Lattice Reduction over Imaginary Quadratic Fields with Applications to Compute-and-Forward

Lyu

Porter

Ling

2018

2018 IEEE Information Theory Workshop (ITW)

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Complex bases, along with direct-sums defined by rings of imaginary quadratic integers, induce algebraic lattices.In this work, we study such lattices and their reduction algorithms. First, when the lattice is spanned over a two dimensional basis, we show that the algebraic variant of Gauss's algorithm returns a basis that corresponds to the successive minima of the lattice in polynomial time if the chosen ring is Euclidean. Second, we extend the celebrated Lenstra-Lenstra-Lovász (LLL) reduction from over real bases to over complex bases. Properties and implementations of the algorithm are examined. In particular, satisfying Lovász's condition requires the ring to be Euclidean. Lastly, as an application, we use the algebraic algorithms to find the network coding matrices in compute-and-forward.Such lattice reduction-based approaches have low complexity which is not dictated by the signal-to-noise (SNR) ratio. Moreover, such approaches can not only preserve the degree-of-freedom of computation rates, but ensure the independence in the code space as well. Index Termslattice reduction, Gauss's algorithm, LLL, compute-and-forward. Initially coding lattices from ConstructionA over rational integers Z or Gaussian integers Z[i] are the main enabler in showing the achievable information rate in C&F. Recently, however, the Construction A lattices have been extended to over the ring of Eisenstein integers Z[ω] [4], [5] and other rings of imaginary quadratic integers [6], This work was presented in part at the IEEE Information Theory Workshop 2018, Guangzhou, China, and submitted in part to the IEEE Information Theory Workshop 2019, Visby, Gotland, Sweden. S. Lyu is with the College

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Section: Appendix a Rotations And Quaternionssupporting

confidence: 63%

Section: Numerical Resultsmentioning

confidence: 99%

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Performance Limits of Lattice Reduction over Imaginary Quadratic Fields with Applications to Compute-and-Forward

Lyu

Porter

Ling

2018

2018 IEEE Information Theory Workshop (ITW)

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“…For very large N , lattices which are simultaneously good for AWGN and quantization [16] can be employed to design optimal Λ f and Λ c for both QIM and MD-QIM. For practical small N , nested lattices can be devised from celebrated lowdimensional lattices such as A 2 , D 4 , E 8 etc.…”

Section: B Workout Examplesmentioning

confidence: 99%

Lattice-Based Minimum-Distortion Data Hiding

Lin

Qin

Lyu

et al. 2021

Preprint

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Lattices have been conceived as a powerful tool for data hiding. While conventional studies and applications focus on achieving the optimal robustness versus distortion tradeoff, in some applications such as data hiding in medical/physiological signals, the primary concern is to achieve a minimum amount of distortion to the cover signal. In this paper, we revisit the celebrated quantization index modulation (QIM) scheme and propose a minimum-distortion version of it, referred to as MD-QIM. The crux of MD-QIM is to move the data point to only the boundary of the Voronoi region of the lattice point indexed by a message, which suffices for subsequent correct decoding. At any fixed code rate, the scheme achieves the minimum amount of distortion by sacrificing the robustness to the additive white Gaussian noise (AWGN) attacks. Simulation results confirm that our scheme significantly outperforms QIM in terms of mean square error (MSE), peak signal to noise ratio (PSNR) and percentage residual difference (PRD).

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“…This is, as shown in this article, the case only if at most two sources are considered, the case studied empirically in [6,10]. More recently, the compute-and-forward protocol has been extended to more general rings of algebraic integers [13].…”

mentioning

confidence: 93%

An Approximation of Theta Functions with Applications to Communications

Barreal¹,

Damir²,

Freij-Hollanti³

et al. 2020

SIAM J. Appl. Algebra Geometry

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Computing the theta series of an arbitrary lattice, and more specifically a related quantity known as the flatness factor, has been recently shown to be important for lattice code design in various wireless communication setups. However, the theta series is in general not known in closed form, excluding a small set of very special lattices. In this article, motivated by the practical applications as well as the mathematical problem itself, a simple approximation of the theta series of a lattice is derived. A rigorous analysis of its accuracy is provided. In relation to this, maximum-likelihood decoding in the context of compute-and-forward relaying is studied. Following previous work, it is shown that the related metric can exhibit a flat behavior, which can be characterized by the flatness factor of the decoding function. Contrary to common belief, we note that the decoding metric can be rewritten as a sum over a random lattice only when at most two sources are considered. Using a particular matrix decomposition, a link between the random lattice and the code lattice employed at the transmitter is established, which leads to an explicit criterion for code design, in contrast to implicit criteria derived previously. Finally, candidate lattices are examined with respect to the proposed criterion using the derived theta series approximation.

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Ring Compute-and-Forward Over Block-Fading Channels

Cited by 14 publications

References 51 publications

Performance Limits of Lattice Reduction over Imaginary Quadratic Fields with Applications to Compute-and-Forward

Performance Limits of Lattice Reduction over Imaginary Quadratic Fields with Applications to Compute-and-Forward

Lattice-Based Minimum-Distortion Data Hiding

An Approximation of Theta Functions with Applications to Communications

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