Spatially-Coupled LDPC Codes for Decode-and-Forward Relaying of Two Correlated Sources over the BEC

We present a decode-and-forward transmission scheme based on spatially-coupled low-density parity-check (SC-LDPC) codes for a network consisting of two (possibly correlated) sources, one relay, and one destination. The links between the nodes are modeled as binary erasure channels. Joint source-channel coding with joint channel decoding is used to exploit the correlation. The relay performs network coding. We derive analytical bounds on the achievable rates for the binary erasure time-division multiple-access relay channel with correlated sources. We then design bilayer SC-LDPC codes and analyze their asymptotic performance for this scenario. We prove analytically that the proposed coding scheme achieves the theoretical limit for symmetric channel conditions and uncorrelated sources. Using density evolution, we furthermore demonstrate that our scheme approaches the theoretical limit also for non-symmetric channel conditions and when the sources are correlated, and we observe the threshold saturation effect that is typical for spatially-coupled systems. Finally, we give simulation results for large block lengths, which validate the DE analysis.Comment: IEEE Transactions on Communications, to appea

A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes

Spatially-coupled LDPC codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially-coupled ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of spatially-coupled LDPC codes under practical constraints

Protograph-Based LDPC Code Design for Shaped Bit-Metric Decoding

A protograph-based low-density parity-check (LDPC) code design technique for bandwidth-efficient coded modulation is presented. The approach jointly optimizes the LDPC code node degrees and the mapping of the coded bits to the bit-interleaved coded modulation (BICM) bit-channels. For BICM with uniform input and for BICM with probabilistic shaping, binary-input symmetric-output surrogate channels for the code design are used. The constructed codes for uniform inputs perform as good as the multi-edge type codes of Zhang and Kschischang (2013). For 8-ASK and 64-ASK with probabilistic shaping, codes of rates 2/3 and 5/6 with blocklength 64800 are designed, which operate within 0.63dB and 0.69dB of continuous AWGN capacity for a target frame error rate of 1e-3 at spectral efficiencies of 1.38 and 4.25 bits/channel use, respectively.Comment: 9 pages, 10 figures. arXiv admin note: substantial text overlap with arXiv:1501.0559

On privacy amplification, lossy compression, and their duality to channel coding

We examine the task of privacy amplification from information-theoretic and coding-theoretic points of view. In the former, we give a one-shot characterization of the optimal rate of privacy amplification against classical adversaries in terms of the optimal type-II error in asymmetric hypothesis testing. This formulation can be easily computed to give finite-blocklength bounds and turns out to be equivalent to smooth min-entropy bounds by Renner and Wolf [Asiacrypt 2005] and Watanabe and Hayashi [ISIT 2013], as well as a bound in terms of the $E_\gamma$ divergence by Yang, Schaefer, and Poor [arXiv:1706.03866 [cs.IT]]. In the latter, we show that protocols for privacy amplification based on linear codes can be easily repurposed for channel simulation. Combined with known relations between channel simulation and lossy source coding, this implies that privacy amplification can be understood as a basic primitive for both channel simulation and lossy compression. Applied to symmetric channels or lossy compression settings, our construction leads to proto- cols of optimal rate in the asymptotic i.i.d. limit. Finally, appealing to the notion of channel duality recently detailed by us in [IEEE Trans. Info. Theory 64, 577 (2018)], we show that linear error-correcting codes for symmetric channels with quantum output can be transformed into linear lossy source coding schemes for classical variables arising from the dual channel. This explains a "curious duality" in these problems for the (self-dual) erasure channel observed by Martinian and Yedidia [Allerton 2003; arXiv:cs/0408008] and partly anticipates recent results on optimal lossy compression by polar and low-density generator matrix codes.Comment: v3: updated to include equivalence of the converse bound with smooth entropy formulations. v2: updated to include comparison with the one-shot bounds of arXiv:1706.03866. v1: 11 pages, 4 figure

Universality for Multi-terminal Problems via Spatial Coupling

Consider the problem of designing capacity-achieving codes for multi-terminal communication scenarios. For point-to-point communication problems, one can optimize a single code to approach capacity, but for multi-terminal problems this translates to optimizing a single code to perform well over the entire region of channel parameters. A coding scheme is called universal if it allows reliable communication over the entire achievable region promised by information theory. It was recently shown that terminated low-density parity-check convolutional codes (also known as spatially-coupled low-density parity-check ensembles) have belief-propagation thresholds that approach their maximum a-posteriori thresholds. This phenomenon, called "threshold saturation via spatial-coupling," was proven for binary erasure channels and then for binary memoryless symmetric channels. This approach provides us with a new paradigm for constructing capacity approaching codes. It was also conjectured that the principle of spatial coupling is very general and that the phenomenon of threshold saturation applies to a very broad class of graphical models. In this work, we consider a noisy Slepian-Wolf problem (with erasure and binary symmetric channel correlation models) and the binary-input Gaussian multiple access channel, which deal with correlation between sources and interference at the receiver respectively. We derive an area theorem for the joint decoder and empirically show that threshold saturation occurs for these multi-user scenarios. We also show that the outer bound derived using the area theorem is tight for the erasure Slepian-Wolf problem and that this bound is universal for regular LDPC codes with large left degrees. As a result, we demonstrate near-universal performance for these problems using spatially-coupled coding systems

Finite-Length Scaling of SC-LDPC Codes With a Limited Number of Decoding Iterations

We propose four finite-length scaling laws to predict the frame error rate (FER) performance of spatially-coupled low-density parity-check codes under full belief propagation (BP) decoding with a limit on the number of decoding iterations and a scaling law for sliding window decoding, also with limited iterations. The laws for full BP decoding provide a choice between accuracy and computational complexity; a good balance between them is achieved by the law that models the number of decoded bits after a certain number of BP iterations by a time-integrated Ornstein-Uhlenbeck process. This framework is developed further to model sliding window decoding as a race between the integrated Ornstein-Uhlenbeck process and an absorbing barrier that corresponds to the left boundary of the sliding window. The proposed scaling laws yield accurate FER predictions