Channel polarization on q-ary discrete memoryless channels by arbitrary kernels

Mori, Ryusaburo; Tanaka, Toshiyuki

doi:10.1109/isit.2010.5513568

Cited by 70 publications

(101 citation statements)

References 8 publications

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“…As an example, a 4 × 4 matrix over GF(q = 2 2 ) composed of a generator matrix of a generalized Reed-Solomon code over GF(2 2 ) has been shown to have the error exponent log 24/(4 log 4) ≈ 0.57312, which is larger than the error exponent 0.51828 for the best possible × matrix over GF(2) for all ≤ 16, which was reported in [6]. Detailed arguments of the above results will be presented elsewhere [10].…”

Section: Channel With Q-ary Inputmentioning

confidence: 77%

Properties of a certain stochastic dynamical system, channel polarization, and polar codes

Tanaka

2010

J. Phys.: Conf. Ser.

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Abstract.A new family of codes, called polar codes, has recently been proposed by Arıkan. Polar codes are of theoretical importance because they are provably capacity achieving with low-complexity encoding and decoding. We first discuss basic properties of a certain stochastic dynamical system, on the basis of which properties of channel polarization and polar codes are reviewed, with emphasis on our recent results. IntroductionThe concept "More is different" is relevant not only in statistical physics but also in various other research fields, such as information theory, where one can find several examples in which properties of a system drastically change as the system size becomes large. As an example, channel coding theorem states that there exist error-correcting codes such that their error probabilities are arbitrarily small and yet their coding rates are arbitrarily close to the channel capacity. Although channel coding theorem is one of the most important theorems in information theory, it is also widely understood that channel coding theorem is a theorem of existence, so that one requires different arguments as to how one can construct practical codes with high performance. The problem of constructing codes which have practically low computational complexity in encoding and decoding, as well as good performance in view of channel coding theorem, remains a problem of particular importance in information theory and coding theory.Proposal of turbo codes in 1990s, as well as subsequent rediscovery of low-density parity-check (LDPC) codes, has ignited extensive research activities on the combination of codes with sparse graphical representation and probability-based iterative decoding algorithms. As a consequence, by now we have practical means of how to construct codes with low computational complexity and high performance. However, rigorous results justifying the high performance of LDPC codes are mostly limited to the case with erasure channels, and we have not arrived at a thorough understanding of LDPC codes for other channels.Arıkan [1] recently proposed polar codes as a class of error-correcting codes which is completely different from LDPC codes. Polar codes are theoretically very interesting because it saturates symmetric channel capacity (maximum mutual information when input distribution is assumed uniform, which is equal to the channel capacity for a symmetric channel) in the large codelength limit, whereas computational complexity of encoding and decoding is polynomial in the codelength. In this paper, we review recent developments regarding polar codes, as well as

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Section: Channel With Q-ary Inputmentioning

confidence: 77%

Properties of a certain stochastic dynamical system, channel polarization, and polar codes

Tanaka

2010

J. Phys.: Conf. Ser.

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“…Furthermore, we can also easily generalize the solution based on homophonic coding to non-binary alphabets (the interval algorithm of [59] works directly in the non-binary case). For the channel coding part, recall that in Section III we have converted a non-binary channel into several binary channels by using the chain rule of mutual information (see formula (13)). Here, the same idea can be applied as well.…”

Section: Discussionmentioning

confidence: 99%

“…Channel polarization for q-ary input alphabets is discussed in [12] and more general constructions based on arbitrary kernels are described in [13]. Furthermore, polar codes have been built exploiting various algebraic structures on the input alphabet [14]- [18].…”

Section: A Symmetric Channel Coding: a Reviewmentioning

confidence: 99%

How to Achieve the Capacity of Asymmetric Channels

Mondelli¹,

Hassani²,

Urbanke³

2018

IEEE Trans. Inform. Theory

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We survey coding techniques that enable reliable transmission at rates that approach the capacity of an arbitrary discrete memoryless channel. In particular, we take the point of view of modern coding theory and discuss how recent advances in coding for symmetric channels help provide more efficient solutions for the asymmetric case. We consider, in more detail, three basic coding paradigms.The first one is Gallager's scheme that consists of concatenating a linear code with a non-linear mapping so that the input distribution can be appropriately shaped. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Furthermore, we derive a scaling law between the gap to capacity, the cardinality of the input and output alphabets, and the required size of the mapper.The second one is an integrated scheme in which the code is used both for source coding, in order to create codewords distributed according to the capacity-achieving input 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 paradigm is based on an idea of Böcherer and Mathar, and separates the two tasks of source coding and channel coding by a chaining construction that binds together several codewords. We present conditions for the source code and the channel code, and we describe how to combine any source code with any channel code that fulfill those conditions, in order to provide capacity-achieving schemes for asymmetric channels. In particular, we show that polar codes, spatially coupled codes, and homophonic codes are suitable as basic building blocks of the proposed coding strategy.Rather than focusing on the exact details of the schemes, the purpose of this tutorial is to present different coding techniques that can then be implemented with many variants. There is no absolute winner and, in order to understand the most suitable technique for a specific application scenario, we provide a detailed comparison that takes into account several performance metrics.

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“…2 shows the effect of such replacement in both SC and BP. The figure presents the simulation result for a code of rate 1 2 and length 2 13 , for which 2 9 information bits are randomly selected and replaced by randomly selected fixed bits. However, for a fair comparison, the same set of information and fixed bits are chosen for both decoding schemes.…”

Section: Theoremmentioning

confidence: 99%

“…6(b) shows the simulation results for employing the guessing algorithm in the gaussian channel. The code we are using is of length 2 13 and has a rate of 1 2 . The maximum number of guesses g max is set to 6.…”

Section: B Girth Of Polar Codesmentioning

confidence: 99%

On bit error rate performance of polar codes in finite regime

Eslami

Pishro-Nik

2010

2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Abstract-Polar codes have been recently proposed as the first low complexity class of codes that can provably achieve the capacity of symmetric binary-input memoryless channels. Here, we study the bit error rate performance of finite-length polar codes under Belief Propagation (BP) decoding. We analyze the stopping sets of polar codes and the size of the minimal stopping set, called "stopping distance". Stopping sets, as they contribute to the decoding failure, play an important role in bit error rate and error floor performance of the code. We show that the stopping distance for binary polar codes, if carefully designed, grows as O( √ N ) where N is the code-length. We provide bit error rate (BER) simulations for polar codes over binary erasure and gaussian channels, showing no sign of error floor down to the BERs of 10 −11 . Our simulations asserts that while finite-length polar codes do not perform as good as LDPC codes in terms of bit error rate, they show superior error floor performance. Motivated by good error floor performance, we introduce a modified version of BP decoding employing a guessing algorithm to improve the BER performance of polar codes. Our simulations for this guessing algorithm show two orders of magnitude improvement over simple BP decoding for the binary erasure channel (BEC), and up to 0.3 dB improvement for the gaussian channel at BERs of 10 −6 .

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Channel polarization on q-ary discrete memoryless channels by arbitrary kernels

Cited by 70 publications

References 8 publications

Properties of a certain stochastic dynamical system, channel polarization, and polar codes

Properties of a certain stochastic dynamical system, channel polarization, and polar codes

How to Achieve the Capacity of Asymmetric Channels

On bit error rate performance of polar codes in finite regime

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