Ruediger Urbanke

Threshold Saturation via Spatial Coupling: Why Convolutional LDPC Ensembles Perform So Well over the BEC

Convolutional low-density parity-check (LDPC) ensembles, introduced by Felström and Zigangirov, have excellent thresholds and these thresholds are rapidly increasing functions of the average degree. Several variations on the basic theme have been proposed to date, all of which share the good performance characteristics of convolutional LDPC ensembles. We describe the fundamental mechanism that explains why “convolutional-like” or “spatially coupled” codes perform so well. In essence, the spatial coupling of individual codes increases the belief-propagation (BP) threshold of the new ensemble to its maximum possible value, namely the maximum a posteriori (MAP) threshold of the underlying ensemble. For this reason, we call this phenomenon “threshold saturation.” This gives an entirely new way of approaching capacity. One significant advantage of this construction is that one can create capacity-approaching ensembles with an error correcting radius that is increasing in the blocklength. Although we prove the “threshold saturation” only for a specific ensemble and for the binary erasure channel (BEC), empirically the phenomenon occurs for a wide class of ensembles and channels. More generally, we conjecture that for a large range of graphical systems a similar saturation of the “dynamical” threshold occurs once individual components are coupled sufficiently strongly. This might give rise to improved algorithms and new techniques for analysis.

Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation

We investigate spatially coupled code ensembles. For transmission over the binary erasure channel, it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a priori threshold of the underlying component ensemble. This explains why convolutional LDPC ensembles, originally introduced by Felstrom and Zigangirov, perform so well over this channel. We show that the equivalent result holds true for transmission over general binary-input memoryless output-symmetric channels. More precisely, given a desired error probability and a gap to capacity, we can construct a spatially coupled ensemble that fulfills these constraints universally on this class of channels under belief propagation decoding. In fact, most codes in this ensemble have this property. The quantifier universal refers to the single ensemble/code that is good for all channels but we assume that the channel is known at the receiver. The key technical result is a proof that, under belief-propagation decoding, spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble. We conclude by discussing some interesting open problems.

Finite-Length Scaling for Polar Codes

Consider a binary-input memoryless output-symmetric channel W. Such a channel has a capacity, call it I(W), and for any R <; I(W) and strictly positive constant Pe we know that we can construct a coding scheme that allows transmission at rate R with an error probability not exceeding Pe. Assume now that we let the rate R tend to I(W) and we ask how we have to scale the blocklength N in order to keep the error probability fixed to Pe. We refer to this as the finite-length scaling behavior. This question was addressed by Strassen as well as Polyanskiy, Poor, and Verdu, and the result is that N must grow at least as the square of the reciprocal of I(W) - R. Polar codes are optimal in the sense that they achieve capacity. In this paper, we are asking to what degree they are also optimal in terms of their finite-length behavior. Since the exact scaling behavior depends on the choice of the channel, our objective is to provide scaling laws that hold universally for all binary-input memoryless output-symmetric channels. Our approach is based on analyzing the dynamics of the un-polarized channels. More precisely, we provide bounds on (the exponent of) the number of subchannels whose Bhattacharyya constant falls in a fixed interval [a, b]. Mathematically, this can be stated as bounding the sequence {1/n logPr(Zn ∈ [a, b])}n∈N, where Zn is the Bhattacharyya process. We then use these bounds to derive tradeoffs between the rate and the block-length. The main results of this paper can be summarized as follows. Consider the sum of Bhattacharyya parameters of subchannels chosen (by the polar coding scheme) to transmit information. If we require this sum to be smaller than a given value Pe > 0, then the required block-length N scales in terms of the rate R <; I(W) as N ≥ α/(I(W) - R)μ, where α is a positive constant that depends on Pe and I(W). We show that μ = 3.579 is a valid choice, and we conjecture that indeed the value of μ can be improved to μ = 3.627, the parameter for the binary erasure channel. Also, we show that with the same requirement on the sum of Bhattacharyya parameters, the blocklength scales in terms of the rate like N ≤ β/(I(W) - R)μ̅, where β is a constant that depends on Pe and I(W), and μ̅ = 6.

Coupled graphical models and their thresholds

The excellent performance of convolutional low-density parity-check codes is the result of the spatial coupling of individual underlying codes across a window of growing size, but much smaller than the length of the individual codes. Remarkably, the belief-propagation threshold of the coupled ensemble is boosted to the maximum-a-posteriori one of the individual system. We investigate the generality of this phenomenon beyond coding theory: we couple general graphical models into a one-dimensional chain of large individual systems. For the later we take the Curie-Weiss, random field Curie-Weiss, If-satisfiability, and Q-coloring models. We always find, based on analytical as well as numerical calculations, that the message passing thresholds of the coupled systems come very close to the static ones of the individual models. The remarkable properties of convolutional low-density parity-check codes are a manifestation of this very general phenomenon.

The compound capacity of polar codes

We consider the compound capacity of polar codes under successive cancellation decoding for a collection of binary-input memoryless output-symmetric channels. By deriving a sequence of upper and lower bounds, we show that in general the compound capacity under successive decoding is strictly smaller than the unrestricted compound capacity.

High Symbol Rate Coherent Optical Transmission Systems: 80 and 107 Gbaud

We demonstrate high speed optical transmission systems using digital coherent detection at all-electronically multiplexed symbol rates of 80 and 107 Gbaud. At 107 Gbaud, we demonstrate a single-carrier polarization division multiplexed quadrature phase shift keyed (PDM-QPSK) line rate of 428 Gb/s. At 80 Gbaud, we achieve a single-carrier line rate of 640 Gb/s using PDM 16-ary quadrature amplitude modulation (16-QAM). Using two optical subcarriers, we demonstrate a 1-Tb/s optical interface and conduct long-haul wavelength-division multiplexed (WDM) transmission on a 200-GHz grid over 3200 km of ultra-large effective area fiber.

Universal Polar Codes

Polar codes, invented by Arikan in 2009, are known to achieve the capacity of any binary-input memoryless output-symmetric channel. Further, both the encoding and the decoding can be accomplished in O(N log(N)) real operations, where N is the blocklength. One of the few drawbacks of the original polar code construction is that it is not universal. This means that the code has to be tailored to the channel if we want to transmit close to capacity. We present two “polar-like” schemes that are capable of achieving the compound capacity of the whole class of binaryinput memoryless symmetric channels with low complexity. Roughly speaking, for the first scheme we stack up N polar blocks of length N on top of each other but shift them with respect to each other so that they form a “staircase.” Then by coding across the columns of this staircase with a standard ReedSolomon code, we can achieve the compound capacity using a standard successive decoder to process the rows (the polar codes) and in addition a standard Reed-Solomon erasure decoder to process the columns. Compared to standard polar codes this scheme has essentially the same complexity per bit but a block length which is larger by a factor O(N log2(N)/ϵ). Here N is the required blocklength for a standard polar code to achieve an acceptable block error probability for a single channel at a distance of at most c from capacity. For the second scheme we first show how to construct a true polar code which achieves the compound capacity for a finite number of channels. We achieve this by introducing special “polarization” steps which “align” the good indices for the various channels. We then show how to exploit the compactness of the space of binary-input memoryless output-symmetric channels to reduce the compound capacity problem for this class to a compound capacity problem for a finite set of channels. This scheme is similar in spirit to standard polar codes, but the price for universality is a considerably larger blocklength.

Achieving Marton’s Region for Broadcast Channels Using Polar Codes

This paper presents polar coding schemes for the two-user discrete memoryless broadcast channel (DM-BC) which achieve Marton’s region with both common and private messages. This is the best achievable rate region known to date, and it is tight for all classes of two-user DM-BCs whose capacity regions are known. To accomplish this task, we first construct polar codes for both the superposition as well as binning strategy. By combining these two schemes, we obtain Marton’s region with private messages only. Finally, we show how to handle the case of common information. The proposed coding schemes possess the usual advantages of polar codes, i.e., they have low encoding and decoding complexity and a superpolynomial decay rate of the error probability. We follow the lead of Goela, Abbe, and Gastpar, who recently introduced polar codes emulating the superposition and binning schemes. To align the polar indices, for both schemes, their solution involves some degradedness constraints that are assumed to hold between the auxiliary random variables and channel outputs. To remove these constraints, we consider the transmission of

Chains of mean-field models

k

Threshold Saturation in Spatially Coupled Constraint Satisfaction Problems

blocks and employ a chaining construction that guarantees the proper alignment of the polarized indices. The techniques described in this paper are quite general, and they can be adopted to many other multiterminal scenarios whenever there polar indices need to be aligned.