Arithmetic geometry of compute and forward

Vázquez-Castro, M. A.

doi:10.1109/itw.2014.6970805

Cited by 10 publications

(9 citation statements)

References 15 publications

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“…In [91], CP&F is investigated for the discrete memoryless channels. The lattice codes are considered for the complex modulo arithmetics in [92]. It is shown that only five lattice code families are capable of complex modulo arithmetics over the Euclidean geometry and their coding gains are calculated.…”

Section: Sum Ratesmentioning

confidence: 99%

The Magic of Superposition: A Survey on Simultaneous Transmission Based Wireless Systems

Altun,

Kurt,

Ozdemir

2021

Preprint

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In conventional communication systems, any interference between two communicating points is regarded as unwanted noise since it distorts the received signals. On the other hand, allowing simultaneous transmission and intentionally accepting the interference of signals and even benefiting from it have been considered for a range of wireless applications. As prominent examples, non-orthogonal multiple access (NOMA), joint source-channel coding, and the computation codes are designed to exploit this scenario. They also inspired many other fundamental works from network coding to consensus algorithms. Especially, federated learning is an emerging technology that can be applied to distributed machine learning networks by allowing simultaneous transmission. Although various simultaneous transmission applications exist independently in the literature, their main contributions are all based on the same principle; the superposition property. In this survey, we aim to emphasize the connections between these studies and provide a guide for the readers on the wireless communication techniques that benefit from the superposition of signals. We classify the existing literature depending on their purpose and application area and present their contributions. The survey shows that simultaneous transmission can bring scalability, security, low-latency, lowcomplexity and energy efficiency for certain distributed wireless scenarios which are inevitable with the emerging Internet of things (IoT) applications.

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Section: Sum Ratesmentioning

confidence: 99%

The Magic of Superposition: A Survey on Simultaneous Transmission Based Wireless Systems

Altun,

Kurt,

Ozdemir

2021

Preprint

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“…Next, we consider the case when Y is C, and discuss the case with X i = F 7 as a typical case [15], [17]. Let P a be the Gaussian distribution with the average a and variance v on C. We define the map u from F 7 to C as follows.…”

Section: B Complex Casementioning

confidence: 99%

“…If the secrecy condition is not imposed and the channel is a multiple Gaussian channel, this problem is a simple example of computation-and-forward [10], [11], [12], [13], [14], [15], [16], [17]. Due to the secrecy condition, we cannot realize this task by their simple application.…”

Section: Introductionmentioning

confidence: 99%

Secure Modulo Sum via Multiple Access Channel

Hayashi¹

2018

Preprint

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We discuss secure computation of modular sum when multiple access channel from distinct players A 1 , . . . , A c to a third party (Receiver) is given. Then, we define the secure modulo sum capacity as the supremum of the transmission rate of modulo sum without information leakage of other information. We derive its useful lower bound, which is numerically calculated under a realistic model that can be realizable as a Gaussian multiple access channel.

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Section: Introductionmentioning

confidence: 99%

“…The idea of using different sets of algebraic integers for compute-and-forward was first proposed independently in [9], [10]. In [9], Vazquez-Castro uses finite constellations carved from some rings of imaginary quadratic integers which also form Euclidean domains (hence PIDs) for compute-and-forward. In [10], instead of being confined in Euclidean domains or PIDs, we go beyond PIDs and construct lattices over rings of imaginary quadratic integers for compute-and-forward.…”

Section: Introductionmentioning

confidence: 99%

Adaptive compute-and-forward with lattice codes over algebraic integers

Huang

Narayanan

Wang

2015

2015 IEEE International Symposium on Information Theory (ISIT)

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We consider the compute-and-forward paradigm with limited feedback. Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all principal ideal domains (PID). A novel scheme called adaptive compute-andforward is proposed to exploit the limited feedback about the channel state by working with the best ring of imaginary quadratic integers. This is enabled by generalizing the famous Construction A from PID to other rings of imaginary quadratic integers which may not form PID and by showing such the construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization. Simulation results show that by adaptively choosing the best ring among the considered ones according to the limited feedback, the proposed adaptive computeand-forward provides a better performance than that provided by the conventional compute-and-forward scheme which works over Gaussian or Eisenstein integers solely. Index TermsCompute-and-forward, physical-layer network coding, lattice codes, and algebraic integers. On one hand, since Z[i] and Z[ω] are instances of imaginary quadratic integers, it seems natural to extend the compute-and-forward framework to general rings of imaginary quadratic integers. On the other hand, since the role of the underlying ring of integers can be effectively thought of as being a quantizer of the channel and Z[ω] is already the best quantizer for C where the channel coefficients belong, it seems unnecessary to pursue compute-and-forward over other rings of integers. In this paper, we first seek to better understand the role of rings of algebraic integers in constructing good lattices. We then use an example, namely compute-and-forward with limited feedback, to demonstrate the benefits of performing compute-and-forward over rings of imaginary quadratic integers other than Z[i] and Z [ω].One important difference between a general ring of imaginary quadratic integers and the Gaussian and Eisenstein integers is that the Gaussian integers and Eisenstein integers are not merely rings, they are principal ideal domains (PIDs). Hence, every ideal is generated by a singleton and one can equivalently work with numbers instead of ideals. The constructions of lattices over these two rings [3]- [5] heavily rely on properties of PID. However, a general ring of imaginary quadratic integers is not a PID. Therefore, in

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Arithmetic geometry of compute and forward

Cited by 10 publications

References 15 publications

The Magic of Superposition: A Survey on Simultaneous Transmission Based Wireless Systems

The Magic of Superposition: A Survey on Simultaneous Transmission Based Wireless Systems

Secure Modulo Sum via Multiple Access Channel

Adaptive compute-and-forward with lattice codes over algebraic integers

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