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Publication# The Velocity of the Propagating Wave for Spatially Coupled Systems With Applications to LDPC Codes

Abstract

We consider the dynamics of message passing for spatially coupled codes and, in particular, the set of density evolution equations that tracks the profile of decoding errors along the spatial direction of coupling. It is known that, for suitable boundary conditions and after a transient phase, the error profile exhibits a "solitonic behavior." Namely, a uniquely shaped wavelike solution develops, which propagates with a constant velocity. Under this assumption, we derive an analytical formula for the velocity in the framework of a continuum limit of the spatially coupled system. The general formalism is developed for spatially coupled low-density parity-check codes on general binary memoryless symmetric channels, which form the main systems of interest in this paper. We apply the formula for special channels and illustrate that it matches the direct numerical evaluation of the velocity for a wide range of noise values. A passible application of the velocity formula to the evaluation of finite size scaling law parameters is also discussed. We conduct a similar analysis for general scalar systems and illustrate the findings with applications to compressive sensing and generalized low-density parity-check codes on the binary erasure or binary symmetric channels.

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For the past 70 years or so, coding theorists have been aiming at designing transmission schemes with efficient encoding and decoding algorithms that achieve the capacity of various noisy channels. It was not until the '90s that graph-based codes, such as low-density parity-check (LDPC) codes, and their associated low-complexity iterative decoding algorithms were discovered and studied in depth. Although these schemes are efficient, they are not, in general, capacity-achieving. More specifically, these codes perform well up to some algorithmic threshold on the channel parameter, which is lower than the optimal threshold. The gap between the algorithmic and optimal thresholds was finally closed by spatial coupling. In the context of coding, the belief propagation algorithm on spatially coupled codes yields capacity-achieving low-complexity transmission schemes. The reason behind the optimal performance of spatially coupled codes is

`seeding'' perfect information on the replicas at the boundaries of the coupling chain. This extra information makes decoding easier near the boundaries, and this effect is then propagated into the coupling chain upon iterations of the decoding algorithm. Spatial coupling was also applied to various other problems that are governed by low-complexity message-passing algorithms, such as random constraint satisfaction problems, compressive sensing, and statistical physics. Each system has an associated algorithmic threshold and an optimal threshold. As with coding, once the underlying graphs are spatially coupled, the algorithms for these systems exhibit optimal performance. In this thesis, we analyze the performance of iterative low-complexity message-passing algorithms on general spatially coupled systems, and we specialize our results in coding theory applications. To do this, we express the evolution of the state of the system (along iterations of the algorithm) in a variational form, in terms of the so-called potential functional, in the continuum limit approximation. This thesis consists of two parts. In the first part, we consider the dynamic phase of the message-passing algorithm, in which iterations of the algorithm modify the state of the spatially coupled system. Assuming that the boundaries of the coupled chain are appropriately `

seeded'', we find a closed-form analytical formula for the velocity with which the extra information propagates into the chain. We apply this result to coupled irregular LDPC code-ensembles with transmission over general BMS channels and to coupled general scalar systems. We perform numerical simulations for several applications and show that our formula gives values that match the empirical, observed velocity. This confirms that the continuum limit is an approximation well-suited to the derivation of the formula. In the second part of this thesis, we consider the static phase of the message-passing algorithm, when it can no longer modify the state of the system. We introduce a novel proof technique that employs displacement convexity, a mathematical tool from optimal transport, to prove that the potential functional is strictly displacement convex under an alternative structure in the space of probability measures. We hence establish the uniqueness of the state to which the spatially coupled system converges, and we characterize it. We apply this result to the (l,r)-regular Gallager ensemble with transmission over the BEC and to coupled general scalar systems.The beginning of 21st century provided us with many answers about how to reach the channel capacity. Polarization and spatial coupling are two techniques for achieving the capacity of binary memoryless symmetric channels under low-complexity decoding algorithms. Recent results prove that another way to achieve capacity is via symmetry, which is the case of the Reed-Muller and extended Bose-Chaudhuri-Hocquenghem (BCH) codes. However, this proof holds only for erasure channel and maximum a posteriori decoding, which is computationally intractable for the general channels.In the first part of this thesis, we talk about the performance improvements that an automorphism group of the code brings on board. We propose two decoding algorithms for the Reed-Muller codes, which are invariant under a large group of permutations and are expected to benefit the most. The former is based on plugging the codeword permutations in successive cancellation decoding, and the latter utilizes the code representation as the evaluations of Boolean monomials. However, despite the performance improvements, it is clear that the decoding complexity grows quickly and becomes impractical for moderate-length codes. In the second part of this thesis, we provide an explanation for this observation. We use the Boolean polynomial representation of the code in order to show that polar-like decoding of sufficiently symmetric codes asymptotically needs an exponential complexity. The automorphism groups of the Reed-Muller and eBCH codes limit the efficiency of their polar-like decoding for long codes, hence we either should focus on short lengths or find another way. We demonstrate that asymptotically same restrictions (although with a slower convergence) hold for more relaxed condition that we call partial symmetry. The developed framework also enables us to prove that the automorphism group of polar codes cannot include a large affine subgroup.In the last part of this thesis, we address a completely different problem. A device-independent quantum key distribution (DIQKD) aims to provide private communication between parties and has the security guarantees that come mostly from quantum physics, without making potentially unrealistic assumptions about the nature of the communication devices. After the quantum part of the DIQKD protocol, the parties share a secret key that is not perfectly correlated. In order to synchronize, some information needs to be revealed publicly, which makes this formulation equivalent to the asymmetric Slepian-Wolf problem that can be solved using binary linear error-correction codes. As any amount of the revealed information reduces the key secrecy, the utilized code should operate close to the finite-length limits. The channel in consideration is non-standard and, due to its experimental nature, it can actually slightly differ from the considered models. In order to solve this problem, we designed a simple scheme using universal SC-LDPC codes and used in the first successful experimental demonstration of DIQKD protocol.

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We consider spatially coupled systems governed by a set of scalar density evolution equations. Such equations track the behavior of message-passing algorithms used, for example, in coding, sparse sensing, or constraint-satisfaction problems. Assuming that the "profile" describing the average state of the algorithm exhibits a solitonic wave-like behavior after initial transient iterations, we derive a formula for the propagation velocity of the wave. We illustrate the formula with two applications, namely Generalized LDPC codes and compressive sensing.