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_87: In a recent work, Nazer and Gastpar proposed the compute-and-forward strategy as a physical-layer network coding scheme. They described a code structure based on nested lattices whose algebraic structure makes the scheme reliable and efficient. In a more recent paper, Niesen and Whiting revealed a fundamental limitation of the decoder used by Nazer and Gastpar. In this work, we consider maximum-likelihood decoding of compute-and-forward, aiming to overcome its limitation. By examining the decoding metric from the viewpoint of Gaussian measures over lattices, we present a design criterion of the lattice code based on the flatness factor, and propose a new decoding algorithm based on inhomogeneous Diophantine approximation.
  @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_53: A novel construction of lattices is proposed. This construction can be thought of as Construction A with codes that can be represented as the Cartesian product of L linear codes over Fp1 , . . . ,FpL , respectively; hence, is referred to as the product construction. The existence of a sequence of such lattices that are good for quantization and Poltyrev-good under multistage decoding is shown. This family of lattices is then used to generate a sequence of nested lattice codes which allows one to achieve the same computation rate of Nazer and Gastpar for computeand-forward under multistage decoding, which is referred to as lattice-based multistage compute-and-forward. Motivated by the proposed lattice codes, two families of signal constellations are then proposed for the separation-based compute-and-forward framework proposed by together with a multilevel coding multistage decoding scheme tailored specifically for these constellations. This scheme is termed separation-based multistage compute-and-forward and is shown having a complexity of the channel coding dominated by the greatest common divisor of the constellation size (may not be a prime number) instead of the constellation size itself.
  @cite_84: We consider a class of lattices built using Construction A, where the underlying code is a non-binary spatially-coupled low density parity check code. We refer to these lattices as spatially-coupled LDA (SCLDA) lattices. SCLDA lattices can be constructed over integers, Gaussian integers and Eisenstein integers. We empirically study the performance of SCLDA lattices under belief propagation (BP) decoding. Ignoring the rate loss from termination, simulation results show that the BP thresholds of SCLDA lattices over integers is 0.11 dB (0.34 dB with the rate loss) and the BP thresholds for SCLDA lattices over Eisenstein integers are 0.08 dB from the Poltyrev limit (0.19 dB with the rate loss). Motivated by this result, we use SCLDA lattice codes over Eisenstein integers for implementing a compute-and-forward protocol. For the examples considered in this paper, the thresholds for the proposed lattice codes are within 0.28 dB from the achievable rate of this coding scheme and within 1.06 dB from the achievable computation rate of Nazer and Gastpar's coding scheme in [6] extended to Eisenstein integers.
  @cite_79: The problem of designing physical-layer network coding (PNC) schemes via nested lattices is considered. Building on the compute-and-forward (C&F) relaying strategy of Nazer and Gastpar, who demonstrated its asymptotic gain using information-theoretic tools, an algebraic approach is taken to show its potential in practical, nonasymptotic, settings. A general framework is developed for studying nested-lattice-based PNC schemes-called lattice network coding (LNC) schemes for short-by making a direct connection between C&F and module theory. In particular, a generic LNC scheme is presented that makes no assumptions on the underlying nested lattice code. C&F is reinterpreted in this framework, and several generalized constructions of LNC schemes are given. The generic LNC scheme naturally leads to a linear network coding channel over modules, based on which noncoherent network coding can be achieved. Next, performance complexity tradeoffs of LNC schemes are studied, with a particular focus on hypercube-shaped LNC schemes. The error probability of this class of LNC schemes is largely determined by the minimum intercoset distances of the underlying nested lattice code. Several illustrative hypercube-shaped LNC schemes are designed based on Constructions A and D, showing that nominal coding gains of 3 to 7.5 dB can be obtained with reasonable decoding complexity. Finally, the possibility of decoding multiple linear combinations is considered and related to the shortest independent vectors problem. A notion of dominant solutions is developed together with a suitable lattice-reduction-based algorithm.
  @cite_59: We investigate channel-coded physical-layer network coding in a two-way relaying scenario, where the end nodes A and B choose their symbols, S_A and S_B, from a small non-binary field, F , and adopt a non-binary PSK modulation. The relay then directly decodes the network-coded combination aS_A+bS_B over F from the noisy received superimposed channel-encoded packets. The advantage of working over non-binary fields is that it offers the opportunity to decode according to multiple decoding coefficients (a,b). As only one of the network-coded combinations needs to be successfully decoded, a key advantage is then a reduction in error probability by attempting to decode against all choices of (a,b). In this paper, we compare different mappings between F and the PSK constellation, and prove that many have identical performance in terms of frame error rate (FER). Moreover, we derive a lower bound on the performance of decoding the network-coded combinations. Simulation results show that if we adopt either i) concatenated Reed-Solomon and convolutional coding or ii) low-density parity check codes, our non-binary constellations can outperform the binary case significantly in the sense of minimizing the FER and, in particular, the ternary constellation has the best FER performance among all considered cases.
  @cite_66: 
summary: As shown by Feng al @cite_79 , any compute-and-forward scheme based on nested lattice codes can be connected to network coding over a finite commutative ring. From this algebraic perspective, the compute-and-forward framework of @cite_54 can be viewed as a special case that connects nested lattice codes generated via Construction A to network coding over a prime-sized finite field. Another important special case is the recent work of Tunali al @cite_66 that develops a compute-and-forward scheme based on nested lattices over Eisenstein integers. For complex-valued channels, this scheme can offer higher computation rates on average (e.g., for Rayleigh fading) since the Eisenstein integers are a better covering of the complex plane than the Gaussian integers employed by @cite_54 . Several recent papers have also used the algebraic perspective of @cite_79 to propose practical codes and constellations for compute-and-forward @cite_87 @cite_59 @cite_84 @cite_53 .