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Concept# Reed–Muller code

Summary

Reed–Muller codes are error-correcting codes that are used in wireless communications applications, particularly in deep-space communication. Moreover, the proposed 5G standard relies on the closely related polar codes for error correction in the control channel. Due to their favorable theoretical and mathematical properties, Reed–Muller codes have also been extensively studied in theoretical computer science.
Reed–Muller codes generalize the Reed–Solomon codes and the Walsh–Hadamard code. Reed–Muller codes are linear block codes that are locally testable, locally decodable, and list decodable. These properties make them particularly useful in the design of probabilistically checkable proofs.
Traditional Reed–Muller codes are binary codes, which means that messages and codewords are binary strings. When r and m are integers with 0 ≤ r ≤ m, the Reed–Muller code with parameters r and m is denoted as RM(r, m). When asked to encode a message consisting of k bits, where \text

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COM-404: Information theory and coding

The mathematical principles of communication that govern the compression and transmission of data and the design of efficient methods of doing so.

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.

Seyed Hamed Hassani, Shrinivas Kudekar, Yury Polyanskiy, Rüdiger Urbanke

Consider a binary linear code of length N, minimum distance d(min), transmission over the binary erasure channel with parameter 0 < epsilon < 1 or the binary symmetric channel with parameter 0 < epsilon < 1/2, and block-MAP decoding. It was shown by Tillich and Zemor that in this case the error probability of the block-MAP decoder transitions "quickly" from delta to 1-delta for any delta > 0 if the minimum distance is large. In particular the width of the transition is of order O(1/root d(min)). We strengthen this result by showing that under suitable conditions on the weight distribution of the code, the transition width can be as small as Theta(1/N1/2-kappa), for any kappa > 0, even if the minimum distance of the code is not linear. This condition applies e.g., to Reed-Mueller codes. Since Theta(1/N-1/2) is the smallest transition possible for any code, we speak of "almost" optimal scaling. We emphasize that the width of the transition says nothing about the location of the transition. Therefore this result has no bearing on whether a code is capacity-achieving or not. As a second contribution, we present a new estimate on the derivative of the EXIT function, the proof of which is based on the Blowing-Up Lemma.

The year 2016, in which I am writing these words, marks the centenary of Claude Shannon, the father of information theory. In his landmark 1948 paper "A Mathematical Theory of Communication", Shannon established the largest rate at which reliable communication is possible, and he referred to it as the channel capacity. Since then, researchers have focused on the design of practical coding schemes that could approach such a limit. The road to channel capacity has been almost 70 years long and, after many ideas, occasional detours, and some rediscoveries, it has culminated in the description of low-complexity and provably capacity-achieving coding schemes, namely, polar codes and iterative codes based on sparse graphs. However, next-generation communication systems require an unprecedented performance improvement and the number of transmission settings relevant in applications is rapidly increasing. Hence, although Shannon's limit seems finally close at hand, new challenges are just around the corner. In this thesis, we trace a road that goes from polar to Reed-Muller codes and, by doing so, we investigate three main topics: unified scaling, non-standard channels, and capacity via symmetry. First, we consider unified scaling. A coding scheme is capacity-achieving when, for any rate smaller than capacity, the error probability tends to 0 as the block length becomes increasingly larger. However, the practitioner is often interested in more specific questions such as, "How much do we need to increase the block length in order to halve the gap between rate and capacity?". We focus our analysis on polar codes and develop a unified framework to rigorously analyze the scaling of the main parameters, i.e., block length, rate, error probability, and channel quality. Furthermore, in light of the recent success of a list decoding algorithm for polar codes, we provide scaling results on the performance of list decoders. Next, we deal with non-standard channels. When we say that a coding scheme achieves capacity, we typically consider binary memoryless symmetric channels. However, practical transmission scenarios often involve more complicated settings. For example, the downlink of a cellular system is modeled as a broadcast channel, and the communication on fiber links is inherently asymmetric. We propose provably optimal low-complexity solutions for these settings. In particular, we present a polar coding scheme that achieves the best known rate region for the broadcast channel, and we describe three paradigms to achieve the capacity of asymmetric channels. To do so, we develop general coding "primitives", such as the chaining construction that has already proved to be useful in a variety of communication problems. Finally, we show how to achieve capacity via symmetry. In the early days of coding theory, a popular paradigm consisted in exploiting the structure of algebraic codes to devise practical decoding algorithms. However, proving the optimality of such coding schemes remained an elusive goal. In particular, the conjecture that Reed-Muller codes achieve capacity dates back to the 1960s. We solve this open problem by showing that Reed-Muller codes and, in general, codes with sufficient symmetry are capacity-achieving over erasure channels under optimal MAP decoding. As the proof does not rely on the precise structure of the codes, we are able to show that symmetry alone guarantees optimal performance.