Reed-Muller codes achieve capacity on erasure channels

Kudekar, Shrinivas; Kumar, Smita; Mondelli, Marco; Pfister, Henry D.; Şaşoğlu, Eren; Urbanke, R.

doi:10.1145/2897518.2897584

Cited by 39 publications

(33 citation statements)

References 80 publications

(87 reference statements)

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“…EXIT functions figure prominently in the study of belief propagation decoding [22], as well as in the recent proof by Kudekar et al that Reed-Muller codes achieve capacity on erasure channels [5]. Let us briefly recall their proof; we will then be able to see how Theorem 4 offers a potential route to generalizing the argument for other channels.…”

Section: Theorem 4 For Any Symmetric Cq Channel W With Input Alphabementioning

confidence: 94%

“…Instead, for a length-n code C encoding length-k messages, the corresponding randomness extractor function is given by f px n q "M . As a simple example of the use of Theorem 3, consider the recent result that Reed-Muller codes achieve the capacity of the binary erasure channel [5]. By duality, this implies that Reed-Muller codes can also extract randomness at the optimal rate from the source describing the joint input and output to the BEC.…”

Section: Theorem 3 For |ψY As Inmentioning

confidence: 99%

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Duality of channels and codes

Renes

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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For any given channel W with classical inputs and possibly quantum outputs, a dual classical-input channel W K can be defined by embedding the original into a channel N with quantum inputs and outputs. Here we give new uncertainty relations for a general class of entropies that lead to very close relationships between the original channel and its dual. Moreover, we show that channel duality can be combined with duality of linear codes, whereupon the uncertainty relations imply that the performance of a given code over a given channel is entirely characterized by the performance of the dual code on the dual channel. This has several applications. In the context of polar codes, it implies that the rates of polarization to ideal and useless channels must be identical. Duality also relates the tasks of channel coding and privacy amplification, implying that the finite blocklength performance of extractors and codes is precisely linked, and that optimal rate extractors can be transformed into capacity-achieving codes, and vice versa. Finally, duality also extends to the EXIT function of any channel and code. Here it implies that for any channel family, if the EXIT function for a fixed code has a sharp transition, then it must be such that the rate of the code equals the capacity at the transition. This gives a different route to proving a code family achieves capacity by establishing sharp EXIT function transitions. IntroductionDuality is an important concept in many branches of mathematics, often enabling given problems to be transformed into dual versions that are simpler to solve. Recently, the author and collaborators have introduced a dual channel in the context of quantum information processing and polar coding [1][2][3][4]. The dual construction applies to channels with classical inputs and classical or quantum outputs and is designed so that the original channel and its dual can both be embedded into the same quantum channel. Constraints on the form of quantum channels then lead to nontrivial constraints on the behavior of the channel and its dual.Here we investigate the notion of duality more comprehensively. We find that it is entirely compatible with the duality of linear codes generally, as well as the notion of channel convolution appearing in belief propagation decoding and polar coding more specifically. Entropic uncertainty relations imply constraints between a wide variety of entropic functions of the channel and code, including EXIT functions. As the class of entropies is quite large, including Rényi entropies for instance, this essentially means that the behavior of a code over a channel is determined by that of the dual code over the dual channel.Channel duality has several applications, which we briefly describe here by way of outlining the structure of the paper. In the next section we set the mathematical stage and define the class of entropies under consideration. Section 3 is then concerned with the definition and properties of dual channels themselves. In particular, duals of simple classica...

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Section: Theorem 4 For Any Symmetric Cq Channel W With Input Alphabementioning

confidence: 94%

Section: Theorem 3 For |ψY As Inmentioning

confidence: 99%

Duality of channels and codes

Renes

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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“…Although Abbe et al did not provide an efficient decoding algorithm, the algorithm we present directly applies here (Section 3.2). The abstraction of the "error-locating pairs" method presented in Section 3 should hopefully be applicable in other contexts too, especially considering the generality of the results of [KP15,KMŞU15].…”

Section: Our Contributionsmentioning

confidence: 99%

“…The results above are obtained by applying recent results of Abbe, Shpilka and Wigderson (STOC, 2015), Kumar and Pfister (2015) and Kudekar et al (2015) regarding the ability of Reed-Muller codes to correct random erasures.…”

Section: Introductionmentioning

confidence: 98%

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Efficiently decoding Reed-Muller codes from random errors

Saptharishi

Shpilka

Volk

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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Reed-Muller codes encode an m-variate polynomial of degree r by evaluating it on all points in {0, 1} m . We denote this code by RM (m, r). The minimal distance of RM(m, r) is 2 m−r and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result.In this work we give an efficient algorithm (in the block length n = 2 m ) for decoding random errors in Reed-Muller codes far beyond the minimal distance. Specifically, for low rate codes (of degree r = o( √ m)) we can correct a random set of (1/2 − o(1))n errors with high probability. For high rate codes (of degree m − r for r = o( m/ log m)), we can correct roughly m r/2 errors.More generally, for any integer r, our algorithm can correct any error pattern in RM(m, m − (2r + 2)) for which the same erasure pattern can be corrected in RM(m, m − (r + 1)). The results above are obtained by applying recent results of Abbe, Shpilka and Wigderson (STOC, 2015), Kumar and Pfister (2015) and Kudekar et al. (2015) regarding the ability of Reed-Muller codes to correct random erasures.The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding Reed-Muller codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the famous Berlekamp-Welch algorithm for decoding Reed-Solomon codes.

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