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.
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 .
@cite_87 @cite_54 @cite_53 @cite_84 @cite_79 @cite_59 @cite_66
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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. In this paper, we consider the use of lattice codes over Eisenstein integers for implementing a compute-and-forward protocol in wireless networks when channel state information is not available at the transmitter. We extend the compute-and-forward paradigm of Nazer and Gastpar to decoding Eisenstein integer combinations of transmitted messages at relays by proving the existence of a sequence of pairs of nested lattices over Eisenstein integers in which the coarse lattice is good for covering and the fine lattice can achieve the Poltyrev limit. Using this result, we show that both the outage performance and error-correcting performance of the nested lattice codebooks over Eisenstein integers surpass those of lattice codebooks over integers considered by Nazer and Gastpar with no additional computational complexity. 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. 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. We address an open question, regarding whether a lattice code with lattice decoding (as opposed to maximum-likelihood (ML) decoding) can achieve the additive white Gaussian noise (AWGN) channel capacity. We first demonstrate how minimum mean-square error (MMSE) scaling along with dithering (lattice randomization) techniques can transform the power-constrained AWGN channel into a modulo-lattice additive noise channel, whose effective noise is reduced by a factor of spl radic (1+SNR SNR). For the resulting channel, a uniform input maximizes mutual information, which in the limit of large lattice dimension becomes 1 2 log (1+SNR), i.e., the full capacity of the original power constrained AWGN channel. We then show that capacity may also be achieved using nested lattice codes, the coarse lattice serving for shaping via the modulo-lattice transformation, the fine lattice for channel coding. We show that such pairs exist for any desired nesting ratio, i.e., for any signal-to-noise ratio (SNR). Furthermore, for the modulo-lattice additive noise channel lattice decoding is optimal. Finally, we show that the error exponent of the proposed scheme is lower bounded by the Poltyrev exponent. Lattice codes used under the Compute-and-Forward paradigm suggest an alternative strategy for the standard Gaussian multiple-access channel (MAC): The receiver successively decodes integer linear combinations of the messages until it can invert and recover all messages. In this paper, this strategy is developed and analyzed. For the two-user MAC, it is shown that without time-sharing, the entire capacity region can be attained with a single-user decoder as soon as the signal-to-noise ratios are above @math . A partial analysis is given for more than two users. Lastly the strategy is extended to the so-called dirty MAC where two interfering signals are known non-causally to the two transmitters in a distributed fashion. Our scheme extends the previously known results and gives new achievable rate regions. In this paper, we propose a new class of lattices constructed from polar codes, namely polar lattices, to achieve the capacity @math of the additive white Gaussian-noise (AWGN) channel for any signal-to-noise ratio (SNR). Our construction follows the multilevel approach of , where on each level we construct a capacity-achieving polar code. The component polar codes are naturally nested, thereby fulfilling the requirement of the multilevel lattice construction. We prove that polar lattices are AWGN-good, in the sense that the error probability (for infinite lattice decoding) vanishes for any fixed volume-to-noise ratio (VNR) greater than @math . Furthermore, using the technique of source polarization, we propose discrete Gaussian shaping over the polar lattice to satisfy the power constraint. The proposed polar lattices permit low-complexity multistage successive cancellation decoding. Both the construction and shaping are explicit, and the overall complexity of encoding and decoding is @math for any fixed target error probability.