We examine the benefits of user cooperation under compute-and-forward. Much like in network coding, receivers in a compute-and-forward network recover finite-field linear combinations of transmitters' messages. Recovery is enabled by linear codes: transmitters map messages to a linear codebook, and receivers attempt to decode the incoming superposition of signals to an integer combination of codewords. However, the achievable computation rates are low if channel gains do not correspond to a suitable linear combination. In response to this challenge, we propose a cooperative approach to compute-and-forward. We devise a lattice-coding approach to block Markov encoding with which we construct a decode-and-forward style computation strategy. Transmitters broadcast lattice codewords, decode each other's messages, and then cooperatively transmit resolution information to aid receivers in decoding the integer combinations. Using our strategy, we show that cooperation offers a significant improvement both in the achievable computation rate and in the diversity-multiplexing tradeoff.
Lattice codes play a fundamental role in compute-and-forward. Early works on lattice codes @cite_7 @cite_21 @cite_36 showed that they are sufficient to achieve capacity for the point-to-point AWGN channel. The performance of lattice codes under lattice decoding ---in which the receiver quantizes the incoming signal to the nearest lattice point---was studied in @cite_15 , and it was shown in @cite_10 that lattice decoding achieves capacity. In addition to compute-and-forward, lattice codes have seen use in a variety of information-theoretic problems, including source coding @cite_4 @cite_37 @cite_32 , physical-layer security @cite_23 @cite_57 @cite_41 , and relay networks @cite_25 @cite_54 @cite_8 @cite_28 .
@cite_37 @cite_4 @cite_7 @cite_8 @cite_36 @cite_41 @cite_54 @cite_21 @cite_28 @cite_32 @cite_57 @cite_23 @cite_15 @cite_10 @cite_25
1512015702 2101872481 2098653764 2400872280
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. It has been conjectured that lattice codes are good for (almost) everything. As an additional bit of evidence for this claim, we offer a few results showing the utility of lattice codes for the AWGN relay channel. We show that the decode-and-forward rates of the relay channel can be achieved using lattice encoding and decoding. We present an encoding decoding technique that uses a doubly-nested lattice code. Encoding is accomplished using a combination of superposition encoding and block Markov encoding, while decoding is accomplished using a strategy reminiscent of Cover and El Gamal's list decoding. Our technique can be extended to a wide variety of relay topologies, including the half-duplex relay channel and the cooperative multiple-access channel. 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. 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.