Reed-Muller codes polarize

Emmanuel Abbé, Min Ye
IEEE COMPUTER SOC, 2019
Conference paper

Abstract

Reed-Muller (RM) codes were introduced in 1954 and have long been conjectured to achieve Shannon's capacity on symmetric channels. The activity on this conjecture has recently been revived with the emergence of polar codes. RM codes and polar codes are generated by the same matrix Gm = [1?] "' but using different subset of rows. RM codes select simply rows having largest weights. Polar codes select instead rows having the largest conditional mutual information proceeding top to down in Gm; while this is a more elaborate and channel-dependent rule, the top-to-down ordering allows Ankan to show that the conditional mutual information polarizes, and this gives directly a capacity-achieving code on any symmetric channel. RM codes are yet to be proved to have such a property, despite the recent success for the erasure channel. In this paper, we connect RM codes to polarization theory. We show that proceeding in the RM code ordering, i.e., not top-to-down but from the lightest to the heaviest rows in Gm, the conditional mutual information again polarizes. We further demonstrate that it does so faster than for polar codes. This implies that Gm contains another code, different than the polar code and called here the twin-RM code, that is provably capacityachieving on any symmetric channel. This gives in particular a necessary condition for RM codes to achieve capacity on symmetric channels. It further gives a sufficient condition if the rows with largest conditional mutual information correspond to the heaviest rows, i.e., if the twin-RM code is the RM code. We demonstrate here that the two codes are at least similar and give further evidence that they are indeed the same.

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Emmanuel Abbé, Min Ye
IEEE COMPUTER SOC, 2019
Conference paper

Abstract

Official source

https://infoscience.epfl.ch/record/276010?ln=en

About this result

Related concepts (10)

Related concepts

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Symmetry in design and decoding of polar-like codes

Kirill Ivanov

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.

EPFL2022

Linear Universal Decoder for Compound Discrete Memoryless Channels

Rethnakaran Pulikkoonattu

Shannon in his seminal work \cite{paper:shannon} formalized the framework on the problem of digital communication of information and storage. He quantified the fundamental limits of compression and transmission rates. The quantity \textit{channel capacity} was coined to give an operational meaning on the ultimate limit of information transfer between two entities, transmitter and receiver. By using a random coding argument, he showed the achievability of reliable communication on any noisy channel as long as the rate is less than the channel capacity. More precisely, he argued that, using a fixed composition random codebook at the sender and a maximum likelihood decoder in the receiver, one can achieve almost zero error communication of information. Shannon's result was a promise on the achievability on the amount of data one can transfer over a medium. Achieving those promises has spurred scientific interest since 1948. Significant effort has been spent to find practical codes which gets close to the Shannon limits. For some class of channels, codes which are capacity achieving has been found since then. Low Density Parity Check Codes (LDPC) is one such capacity achieving family of codes for the class of binary erasure channels. Recently, Arikan proposed a family of codes known as Polar codes which are found to be capacity achieving for binary input memoryless channels. Slowly but steadily the longstanding problem of achieving Shannon limit is thus getting settled. In order to realize a practical system which can achieve the Shannon limits, two important things are necessary. One of them is an efficient capacity achieving code sequence and the other is an implementable decoding rule. Both these require the knowledge of the probabilistic law of the channel through which communication take place. 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This is the central theme in what is known\footnote{Also referred as \textit{coding for class of channels} as discussed in \cite{paper:blackwell}, \cite{book:Wolfowitz}.} as \textit{coding for compound channels}\cite{book:Csiszar}. Designing a reliable communication system for compound channels essentially involve building a codebook at the transmitter and a decoding rule at the receiver. The encoder-decoder pair together must guarantee reliable communication over a set of channels. In the general setting, no feedback is assumed from the receiver to the transmitter. Thus, a single coding strategy (codebook) must be devised (and fixed) upfront prior to transmission. At the receiver end, the decoding strategy must be devised independent of any knowledge of the channel. Of course, the codebook devised at the sender is perfectly known to the receiver. The answer to the question on whether such a communication system can be build is on the affirmative. In order to talk about the existence of reliable communication strategies, one must first talk about the maximum possible rate, the capacity. The highest achievable rate in a compound setting is known as the \textit{compound capacity} or capacity of the family. This is analogous to the famous Shannon capacity being the ultimate limit on achievable rate for any single channel. The capacity of compound set of memoryless channels has been studied by Blackwell et al in \cite{paper:blackwell} and that of linear dispersive channels is investigated by Root and Varaiya in \cite{paper:root:varaiya}. Lapidoth and Telatar looked at the compound capacity for families of channels with memory, more specifically the finite state channels \cite{paper:lapidoth-telatar1998}. As a special case, they have also derived the compound channel capacity of a class of Gilbert-Elliot channels. A decoder which operates without the knowledge of the channel in this setup is called a universal decoder. It is known that Maximum Mutual Information (MMI) decoders proposed by Goppa are universal. MMI decoders compute the empirical mutual between a received codeword against the codebook and find the best matching word as the true estimate. The complexity of MMI decoder remain fixed even if we were to find structured codes. This motivate us to ask the question whether we can build a universal decoder which offer better structure. Decoding rules which brings additive nature were considered in the literature as a potential scheme. Our work in this has been driven by this line of thoughts. In this work, we focus on class of discrete memoryless channels (DMC) and more specifically binary memoryless channels. We show that it is possible to build a linear universal decoder for any compound binary memoryless channels. The recent introduction of Polar codes motivated us to look into their suitability to the compound channel setup. We have carried out a preliminary investigation in this direction. While it is not clear whether Polar codes are universal under the optimum universal decoding, we find that they are universal for certain restricted classes of compound BMC sets.

2009

The general subject considered in this thesis is a recently discovered coding technique, polar coding, which is used to construct a class of error correction codes with unique properties. In his ground-breaking work, Arikan proved that this class of codes, called polar codes, achieve the symmetric capacity --- the mutual information evaluated at the uniform input distribution ---of any stationary binary discrete memoryless channel with low complexity encoders and decoders requiring in the order of

O(N\log N)

operations in the block-length

N

. This discovery settled the long standing open problem left by Shannon of finding low complexity codes achieving the channel capacity. Polar codes are not only appealing for being the first to 'close the deal'. In contrast to most of the existing coding schemes, polar codes admit an explicit low complexity construction. In addition, for symmetric channels, the polar code construction is deterministic; the theoretically beautiful but practically limited "average performance of an ensemble of codes is good, so there must exist one particular code in the ensemble at least as good as the average'' formalism of information theory is bypassed. Simulations are thus not necessary in principle for evaluating the error probability which is shown in a study by Telatar and Arikan to scale exponentially in the square root of the block-length. As such, at the time of this writing, polar codes are appealing for being the only class of codes proved, and proved with mathematical elegance, to possess all of these properties. Polar coding settled an open problem in information theory, yet opened plenty of challenging problems that need to be addressed. This novel coding scheme is a promising method from which, in addition to data transmission, problems such as data compression or compressed sensing, which includes all types of measurement processes like the MRI or ultrasound, could benefit in terms of efficiency. To make this technique fulfill its promise, the original theory has been, and should still be, extended in multiple directions. A significant part of this thesis is dedicated to advancing the knowledge about this technique in two directions. The first one provides a better understanding of polar coding by generalizing some of the existing results and discussing their implications, and the second one studies the robustness of the theory over communication models introducing various forms of uncertainty or variations into the probabilistic model of the channel. See the fulltext of the thesis for the complete abstract.

EPFL2015