S. Hamed Hassani

On the construction of polar codes

We consider the problem of efficiently constructing polar codes over binary memoryless symmetric (BMS) channels. The complexity of designing polar codes via an exact evaluation of the polarized channels to find which ones are “good” appears to be exponential in the block length. In [3], Tal and Vardy show that if instead the evaluation if performed approximately, the construction has only linear complexity. In this paper, we follow this approach and present a framework where the algorithms of [3] and new related algorithms can be analyzed for complexity and accuracy. We provide numerical and analytical results on the efficiency of such algorithms, in particular we show that one can find all the “good” channels (except a vanishing fraction) with almost linear complexity in block-length (except a polylogarithmic factor).

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.

On the scaling of polar codes: II. The behavior of un-polarized channels

We consider the asymptotic behavior of the polarization process for polar codes when the blocklength tends to infinity. In particular, we study the asymptotics of the cumulative distribution ℙ(Z<inf>n</inf> ≤ z), where Z<inf>n</inf> = Z(W<inf>n</inf>) is the Bhat-tacharyya process, and its dependence on the rate of transmission R. We show that for a BMS channel W, for R < I(W) we have lim<inf>n→8</inf> ℙ equations R and for R < 1 − I(W) we have <inf>n→8</inf> ℙ equations R, where Q(x) is the probability that a standard normal random variable exceeds x. As a result, if we denote by ℙ^SC<inf>e</inf> (n,R) the probability of error using polar codes of block-length N = 2ⁿ and rate R < I(W) under successive cancellation decoding, then log(−log(ℙ^SC<inf>e</inf> (n,R))) scales as equations. We also prove that the same result holds for the block error probability using the MAP decoder, i.e., for log(−log(ℙ^MAP<inf>e</inf> (n,R))).

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.

How to achieve the capacity of asymmetric channels

We describe coding techniques that achieve the capacity of a discrete memoryless asymmetric channel. To do so, we discuss how recent advances in coding for symmetric channels yield more efficient solutions also for the asymmetric case. In more detail, we consider three basic approaches. The first one is Gallagers scheme that concatenates a linear code with a non-linear mapper, in order to bias the input distribution. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Further, we derive a scaling law between the gap to capacity, the cardinality of channel input and output alphabets, and the required size of the mapper. The second one is an integrated approach in which the coding scheme is used both for source coding, in order to create codewords with the capacity-achieving distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. The third approach is based on an idea due to Böcherer and Mathar and separates completely the two tasks of source coding and channel coding by “chaining” together several codewords. We prove that we can combine any suitable source code with any suitable channel code in order to provide optimal schemes for asymmetric channels. In particular, polar codes and spatially coupled codes fulfill the required conditions.

Chains of mean-field models

We consider a collection of Curie-Weiss (CW) spin systems, possibly with a random field, each of which is placed along the positions of a one-dimensional chain. The CW systems are coupled together by a Kac-type interaction in the longitudinal direction of the chain and by an infinite-range interaction in the direction transverse to the chain. Our motivations for studying this model come from recent findings in the theory of error-correcting codes based on spatially coupled graphs. We find that, although much simpler than the codes, the model studied here already displays similar behavior. We are interested in the van der Waals curve in a regime where the size of each Curie-Weiss model tends to infinity, and the length of the chain and range of the Kac interaction are large but finite. Below the critical temperature, and with appropriate boundary conditions, there appears a series of equilibrium states representing kink-like interfaces between the two equilibrium states of the individual system. The van der Waals curve oscillates periodically around the Maxwell plateau. These oscillations have a period inversely proportional to the chain length and an amplitude exponentially small in the range of the interaction; in other words, the spinodal points of the chain model lie exponentially close to the phase transition threshold. The amplitude of the oscillations is closely related to a Peierls-Nabarro free energy barrier for the motion of the kink along the chain. Analogies to similar phenomena and their possible algorithmic significance for graphical models of interest in coding theory and theoretical computer science are pointed out.

Threshold Saturation in Spatially Coupled Constraint Satisfaction Problems

We consider chains of random constraint satisfaction models that are spatially coupled across a finite window along the chain direction. We investigate their phase diagram at zero temperature using the survey propagation formalism and the interpolation method. We prove that the SAT-UNSAT phase transition threshold of an infinite chain is identical to the one of the individual standard model, and is therefore not affected by spatial coupling. We compute the survey propagation complexity using population dynamics as well as large degree approximations, and determine the survey propagation threshold. We find that a clustering phase survives coupling. However, as one increases the range of the coupling window, the survey propagation threshold increases and saturates towards the phase transition threshold. We also briefly discuss other aspects of the problem. Namely, the condensation threshold is not affected by coupling, but the dynamic threshold displays saturation towards the condensation one. All these features may provide a new avenue for obtaining better provable algorithmic lower bounds on phase transition thresholds of the individual standard model.

Universal bounds on the scaling behavior of polar codes

We consider the problem of determining the tradeoff between the rate and the block-length of polar codes for a given block error probability when we use the successive cancellation decoder. We take the sum of the Bhattacharyya parameters as a proxy for the block error probability, and show that there exists a universal parameter μ such that for any binary memoryless symmetric channel W with capacity I(W), reliable communication requires rates that satisfy R <; I(W) - αN-1/μ, where α is a positive constant and N is the block-length. We provide lower bounds on μ, namely μ ≥ 3.553, and we conjecture that indeed μ = 3.627, the parameter for the binary erasure channel.

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 Martons 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 Martons 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 k 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.