Weight Distribution and List-Decoding Size of Reed–Muller Codes

Kaufman, Tali; Lovett, Shachar; Porat, Ely

doi:10.1109/tit.2012.2184841

Cited by 51 publications

(108 citation statements)

References 19 publications

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“…These are naturally important by themselves, and, furthermore, tight weight distribution bounds turns out to be crucial for achieving capacity for the BEC in Theorem 1.1 above, as well as for achieving capacity for the BSC in Theorem 1.7 below. Our bound extends an important recent result of Kaufman, Lovett and Porat on the weight-distribution of Reed-Muller codes [KLP12], using a simple variant of their technique. Kaufman et al gave a bound that was tight for r = O(1), but degrades as r grows.…”

Section: Weight Distribution and List Decodingsupporting

confidence: 78%

“…As in the paper of [KLP12], almost the exact same proof as our proof of Theorem 1.5 yields a bound for list-decoding of Reed-Muller codes, for which we get similar improvements. Following [KLP12] we denote:…”

Section: Weight Distribution and List Decodingsupporting

confidence: 65%

“…Amazingly enough, very little was known about the weight distribution of Reed-Muller code until the recent breakthrough paper of [KLP12], who gave nearly tight bounds for constant degree polynomials for both. The results of [KLP12] also apply to list-decoding of RM codes, which was previously investigated in [GKZ08]. We need a sharpening of their upper bound for two of our results, which we prove by refining their method.…”

Section: For the Case Of The Bec [Lmsmentioning

confidence: 99%

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Reed–Muller Codes for Random Erasures and Errors

Abbé

Shpilka

Wigderson

2015

IEEE Trans. Inform. Theory

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This paper studies the parameters for which Reed-Muller (RM) codes over GF (2) can correct random erasures and random errors with high probability, and in particular when can they achieve capacity for these two classical channels. Necessarily, the paper also studies properties of evaluations of multi-variate GF (2) polynomials on random sets of inputs.For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about square root of the number of errors at capacity.The proofs of these four results are based on different techniques, which we find interesting in their own right. In particular, we study the following questions about E(m, r), the matrix whose rows are truth tables of all monomials of degree ≤ r in m variables. What is the most (resp. least) number of random columns in E(m, r) that define a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees r, which we use to show that RM codes achieve capacity for erasures in these regimes.Our decoding from random errors follows from the following novel reduction. For every linear code C of sufficiently high rate we construct a new code C , also of very high rate, such that for every subset S of coordinates, if C can recover from erasures in S, then C can recover from errors in S. Specializing this to RM codes and using our results for erasures imply our result on unique decoding of RM codes at high rate.Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent [KLP12] bounds from constant degree to linear degree polynomials.

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Section: Weight Distribution and List Decodingsupporting

confidence: 78%

Section: Weight Distribution and List Decodingsupporting

confidence: 65%

Section: For the Case Of The Bec [Lmsmentioning

confidence: 99%

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Reed–Muller Codes for Random Erasures and Errors

Abbé

Shpilka

Wigderson

2015

IEEE Trans. Inform. Theory

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“…The early work in [30]- [32] culminated in obtaining asymptotically tight bounds (fixed order v and asymptotic n) for their weight distribution [33]. Also, there is considerable interest in constructing low-complexity decoding algorithms, see [34], [35] and a series of papers by Dumer et al [36]- [38].…”

Section: B Reed-muller Codesmentioning

confidence: 99%

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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“…Kaufman, Lovett and Porat [KLP12] later improved this global list-decoder up to distance 2 * δ d = δ d−1 , twice the list-decoding radius of RM(n, d).…”

Section: Introductionmentioning

confidence: 99%

Approximate Local Decoding of Cubic Reed-Muller Codes Beyond the List Decoding Radius

Hatami

Tulsiani

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider the question of decoding Reed-Muller codes over F n 2 beyond their list-decoding radius. Since, by definition, in this regime one cannot demand an efficient exact listdecoder, we seek an approximate decoder: Given a word F and radii r > r > 0, the goal is to output a codeword within radius r of F , if there exists a codeword within distance r. As opposed to the list decoding problem, it suffices here to output any codeword with this property, since the list may be too large if r exceeds the list decoding radius.Prior to our work, such decoders were known for ReedMuller codes of degree 2, due to works of Wolf and the second author [FOCS 2011]. In this work we make the first progress on this problem for the degree 3 where the list decoding radius is 1/8. We show that there is a constant δ = 1/2− 1/8 > 1/8 and an efficient approximate decoder, that given query access to a function F : F n 2 → F 2 , such that F is within distance r = δ − ε from a cubic polynomial, runs in time polynomial in message length and outputs with high probability a cubic polynomial which is at distance at most r = 1/2 − ε from F , where ε is a quasi polynomial function of ε.

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Weight Distribution and List-Decoding Size of Reed–Muller Codes

Cited by 51 publications

References 19 publications

Reed–Muller Codes for Random Erasures and Errors

Reed–Muller Codes for Random Erasures and Errors

Reed-Muller codes achieve capacity on erasure channels

Approximate Local Decoding of Cubic Reed-Muller Codes Beyond the List Decoding Radius

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