text: The compute-and-forward framework permits each receiver in a Gaussian network to directly decode a linear combination of the transmitted messages. The resulting linear combinations can then be employed as an end-to-end communication strategy for relaying, interference alignment, and other applications. Recent efforts have demonstrated the advantages of employing unequal powers at the transmitters and decoding more than one linear combination at each receiver. However, neither of these techniques fit naturally within the original formulation of compute-and-forward. This paper proposes an expanded compute-and-forward framework that incorporates both of these possibilities and permits an intuitive interpretation in terms of signal levels. Within this framework, recent achievability and optimality results are unified and generalized.
  @cite_22: We consider a communication system where two transmitters wish to exchange information through a central relay. The transmitter and relay nodes exchange data over synchronized, average power constrained additive white Gaussian noise channels with a real input with signal-to-noise ratio (SNR) of snr. An upper bound on the capacity is 1 2 log(1 + snr) bits per transmitter per use of the multiple access phase and broadcast phase of the bidirectional relay channel. We show that, using lattice codes and lattice decoding, we can obtain a rate of 1 2 log(1 2 + snr) bits per transmitter, which is essentially optimal at high SNR. The main idea is to decode the sum of the codewords modulo a lattice at the relay followed by a broadcast phase which performs Slepian-Wolf coding. We also show that if the two transmitters use identical lattices with minimum angle decoding, we can achieve the same rate of 1 2 log(1 2 + snr). The proposed scheme can be thought of as a joint physical-layer network-layer code which outperforms other recently proposed analog network coding schemes.
  @cite_8: We study the problem of multicasting over a network of multiple-access channels (MACs). The separation-based solution to this problem is to reduce each MAC to a set of noiseless bit pipes via a channel code and then employ network coding. Sometimes, however, the physical-layer structure of the MAC can be exploited more advantageously. In many cases of interest, the MAC output is a (deterministic) function of its inputs, corrupted by noise. We develop structured codes to exploit the natural function of a MAC to reliably compute functions as part of a network code and show that in many scenarios of interest our scheme outperforms the separation-based solution. If each MAC can be written as a sum over some finite field plus noise, then our achievable rate coincides with the max-flow min-cut bound.
  @cite_60: In this paper, we consider a class of single-source multicast relay networks. We assume that all outgoing channels of a node in the network to its neighbors are orthogonal while the incoming signals from its neighbors can interfere with each other. We first focus on Gaussian relay networks with interference and find an achievable rate using a lattice coding scheme. We show that the achievable rate of our scheme is within a constant bit gap from the information theoretic cut-set bound, where the constant depends only on the network topology, but not on the transmit power, noise variance, and channel gains. This is similar to a recent result by Avestimehr, Diggavi, and Tse, who showed an approximate capacity characterization for general Gaussian relay networks. However, our achievability uses a structured code instead of a random one. Using the idea used in the Gaussian case, we also consider a linear finite-field symmetric network with interference and characterize its capacity using a linear coding scheme.
  @cite_54: Interference is usually viewed as an obstacle to communication in wireless networks. This paper proposes a new strategy, compute-and-forward, that exploits interference to obtain significantly higher rates between users in a network. The key idea is that relays should decode linear functions of transmitted messages according to their observed channel coefficients rather than ignoring the interference as noise. After decoding these linear equations, the relays simply send them towards the destinations, which given enough equations, can recover their desired messages. The underlying codes are based on nested lattices whose algebraic structure ensures that integer combinations of codewords can be decoded reliably. Encoders map messages from a finite field to a lattice and decoders recover equations of lattice points which are then mapped back to equations over the finite field. This scheme is applicable even if the transmitters lack channel state information.
  @cite_43: We take a new look at the issue of network capacity. It is shown that network coding is an essential ingredient in achieving the capacity of a network. Building on recent work by (see Proc. 2001 IEEE Int. Symp. Information Theory, p.102), who examined the network capacity of multicast networks, we extend the network coding framework to arbitrary networks and robust networking. For networks which are restricted to using linear network codes, we find necessary and sufficient conditions for the feasibility of any given set of connections over a given network. We also consider the problem of network recovery for nonergodic link failures. For the multicast setup we prove that there exist coding strategies that provide maximally robust networks and that do not require adaptation of the network interior to the failure pattern in question. The results are derived for both delay-free networks and networks with delays.
  @cite_72: This paper considers relaying techniques that increase the achievable throughput in multi-hop wireless networks by taking advantage of the bi-directional traffic flow. Such a relaying technique is termed relaying with Bi-directional Amplification of Throughput (BAT-relaying). The BAT-relaying is utilizing the concept of anti-packets, defined for bi-directional traffic flows. The relay node combines the anti-packets that are destined for different nodes and broadcasts the combined packet. Two BAT-relaying techniques have been proposed previously, Decode-and-Forward (DF) BAT-relaying and Amplify-and-Forward (AF) BAT-relaying. While in DF the relay node combines the packets by an XOR operation, AF BAT-relaying utilizes the inherent packet combining provided by the multiple access channel. In an errorless channel, AF has always higher achievable throughput than DF, but in noisy channels the noise amplification can severely degrade the performance of AF. In this paper we introduce a new scheme for BAT-relaying, termed Denoise-And-Forward (DNF) BAT-Relaying. The DNF BAT-relaying also makes use of the combining provided by the multiple access channel, but it removes the noise from the combined anti-packets before broadcasting to the destinations. While in the noiseless channel DNF and AF offer the same throughput performance which is superior to DF BAT-relaying, in large regions of the lower SNR values DNF BAT-relaying has the best throughput performance of all three schemes. Due to the unconventional nature of the BAT-relaying schemes, there are many open issues for further investigation. The design of a practical DNF scheme concerns several protocol layers, including modulation and coding.
  @cite_71: Consider a communication network in which certain source nodes multicast information to other nodes on the network in the multihop fashion where every node can pass on any of its received data to others. We are interested in how fast each node can receive the complete information, or equivalently, what the information rate arriving at each node is. Allowing a node to encode its received data before passing it on, the question involves optimization of the multicast mechanisms at the nodes. Among the simplest coding schemes is linear coding, which regards a block of data as a vector over a certain base field and allows a node to apply a linear transformation to a vector before passing it on. We formulate this multicast problem and prove that linear coding suffices to achieve the optimum, which is the max-flow from the source to each receiving node.
  @cite_86: A main distinguishing feature of a wireless network compared with a wired network is its broadcast nature, in which the signal transmitted by a node may reach several other nodes, and a node may receive signals from several other nodes simultaneously. Rather than a blessing, this feature is treated more as an interference-inducing nuisance in most wireless networks today (e.g., IEEE 802.11). The goal of this paper is to show how the concept of network coding can be applied at the physical layer to turn the broadcast property into a capacity-boosting advantage in wireless ad hoc networks. Specifically, we propose a physical-layer network coding (PNC) scheme to coordinate transmissions among nodes. In contrast to "straightforward" network coding which performs coding arithmetic on digital bit streams after they have been received, PNC makes use of the additive nature of simultaneously arriving electromagnetic (EM) waves for equivalent coding operation. PNC can yield higher capacity than straight-forward network coding when applied to wireless networks. We believe this is a first paper that ventures into EM-wave-based network coding at the physical layer and demonstrates its potential for boosting network capacity. PNC opens up a whole new research area because of its implications and new design requirements for the physical, MAC, and network layers of ad hoc wireless stations. The resolution of the many outstanding but interesting issues in PNC may lead to a revolutionary new paradigm for wireless ad hoc networking.
  @cite_31: We introduce a new class of problems called network information flow which is inspired by computer network applications. Consider a point-to-point communication network on which a number of information sources are to be multicast to certain sets of destinations. We assume that the information sources are mutually independent. The problem is to characterize the admissible coding rate region. This model subsumes all previously studied models along the same line. We study the problem with one information source, and we have obtained a simple characterization of the admissible coding rate region. Our result can be regarded as the max-flow min-cut theorem for network information flow. Contrary to one's intuition, our work reveals that it is in general not optimal to regard the information to be multicast as a "fluid" which can simply be routed or replicated. Rather, by employing coding at the nodes, which we refer to as network coding, bandwidth can in general be saved. This finding may have significant impact on future design of switching systems.
  @cite_17: 
summary: The main concept underlying compute-and-forward is that the superposition property of the wireless medium can be exploited for network coding @cite_31 @cite_71 @cite_43 . This phenomenon was independently and concurrently discovered by @cite_72 @cite_8 @cite_86 , with the latter coining the phrase Subsequent efforts @cite_17 @cite_22 @cite_60 developed lattice coding strategies for communicating the sum of messages to a single receiver. This lead to the compute-and-forward framework @cite_54 for multiple receivers that recover linear combinations of the messages (albeit with equal power constraints, unlike the single receiver framework of @cite_60 ).