Reed–Muller Codes for Random Erasures and Errors

Abbé, Emmanuel; Shpilka, Amir; Wigderson, Avi

doi:10.1109/tit.2015.2462817

Cited by 51 publications

(24 citation statements)

References 41 publications

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“…We note that Pyndiah [48] has proposed a simple soft-input soft-output (approximated MAP) decoder for BCH codes, based on Chase algorithm, that gives excellent BER performance. RM codes achieve capacity on erasure channels and reduce PMEPR [49,50].…”

Section: Numerical Resultsmentioning

confidence: 99%

Coding to reduce the interference to carrier ratio of OFDM signals

Smida

2017

J Wireless Com Network

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In this paper, we propose a simple approach to reduce the sensitivity of OFDM system to the carrier frequency offset (CFO). The main idea is to choose a well-known channel code and to rotate each coordinate by a fixed phase shift such that the maximum inter-carrier interference (ICI) taken over all sub-carriers is minimized. This approach is based on a geometric interpretation of the peak interference to carrier ratio (PICR) of OFDM signals. Simulation results show that a reduction in PICR of 7 dB can be easily achieved. Furthermore, we addressed the fundamental limit of the proposed technique by providing both an exact and an approximate lower bound of PICR of phase-shifted binary codes. The bounds are applied to some codes: non-redundant binary code, BCH codes, and Reed-Muller codes. Simulation results demonstrated that phase-shift designs approach lower bounds and are resilient to CFO change.

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Section: Numerical Resultsmentioning

confidence: 99%

Coding to reduce the interference to carrier ratio of OFDM signals

Smida

2017

J Wireless Com Network

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“…Note that by the Chernoff-Hoeffding bound, (see e.g., [AS92]), the probability that more than pn + ω( √ pn) errors occur in BSC(p) (or BEC(p)) is o(1), and so we can restrict ourselves to the case of a fixed number s of random errors, by setting the corruption probability to be p = s/n. We refer to [ASW15] for further discussion on this subject.…”

Section: Random Errorsmentioning

confidence: 99%

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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“…In [27], a modified construction of polar codes is analyzed and the results again suggest that Reed-Muller codes achieve capacity on the BEC. For rates approaching either 0 or 1 with sufficient speed, it has recently been shown by Abbe et al that Reed-Muller codes can correct almost all erasure patterns up to the capacity limit 2 [28], [29]. Beyond erasure channels, it is conjectured in [25] that the sequence of rate-1/2 selfdual Reed-Muller codes achieves capacity on the binary-input AWGN channel.…”

Section: B Reed-muller Codesmentioning

confidence: 99%

“…In particular, [28,Theorem 1.8] shows that an error pattern can be corrected by RM(n − (2t + 2), n) under block-MAP decoding whenever an erasure pattern with the same support can be corrected by RM(n − (t + 1), n) under block-MAP decoding. Using the algorithm in [88], these error patterns can even be corrected efficiently.…”

Section: Beyond the Erasure Channelmentioning

confidence: 99%

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Reed-Muller codes achieve capacity on erasure channels

Kudekar

Kumar

Mondelli

et al. 2016

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

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Abstract-We introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise structure of the codes our method exploits code symmetry. In particular, the technique applies to any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge, and the permutation group of each code is doubly transitive. In other words, we show that symmetry alone implies near-optimal performance.An important consequence of this result is that a sequence of Reed-Muller codes with increasing blocklength and converging rate achieves capacity. This possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1. Moreover, these results extend naturally to all affine-invariant codes and, thus, to extended primitive narrow-sense BCH codes. This also resolves, in the affirmative, the existence question for capacity-achieving sequences of binary cyclic codes. The primary tools used in the proof are the sharp threshold property for symmetric monotone boolean functions and the area theorem for extrinsic information transfer functions.

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Reed–Muller Codes for Random Erasures and Errors

Cited by 51 publications

References 41 publications

Coding to reduce the interference to carrier ratio of OFDM signals

Coding to reduce the interference to carrier ratio of OFDM signals

Efficiently decoding Reed-Muller codes from random errors

Reed-Muller codes achieve capacity on erasure channels

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