On the Performance of Reed-Muller Codes with respect to Random Errors and Erasures

Sberlo, Ori; Shpilka, Amir

doi:10.48550/arxiv.1811.12447

Cited by 4 publications

(28 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Observe that when the degree r is linear in m Theorem 6 only applies to weights that are some constant smaller than 1/2 and does not give information about the number of polynomials that have bias o(1). Such a result was obtained by Sberlo and Shpilka [43], and it played an important role in their results on the capacity of RM codes. They first proved a result for the case that r < m/2 and then for the general case (with a weaker bound).…”

Section: A Resultssupporting

confidence: 57%

“…IV-A RM codes achieve capacity at low rate [10], [43] RM codes polarize and Twin-RM codes achieve capacity on any BMS [17] . .…”

Section: Capacity Resultsmentioning

confidence: 98%

“…Sberlo and Shpilka [43] polished the techniques of [10] and managed to obtain good estimates for every degree.…”

Section: A Resultsmentioning

confidence: 99%

“…Recently, Samorodnitsky [23] proved a remarkable general result regarding the weight enumerator of codes that either they or their dual code achieve capacity for the BEC. His techniques are completely different than the techniques of [10], [22], [43].…”

Section: A Resultsmentioning

confidence: 99%

See 3 more Smart Citations

Reed-Muller Codes: Theory and Algorithms

Abbé

Shpilka

2020

Preprint

Self Cite

View full text Add to dashboard Cite

Reed-Muller (RM) codes are among the oldest, simplest and perhaps most ubiquitous family of codes. They are used in many areas of coding theory in both electrical engineering and computer science. Yet, many of their important properties are still under investigation. This paper covers some of the recent developments regarding the weight enumerator and the capacity-achieving properties of RM codes, as well as some of the algorithmic developments. In particular, the paper discusses the recent connections established between RM codes, thresholds of Boolean functions, polarization theory, hypercontractivity, and the techniques of approximating low weight codewords using lower degree polynomials. It then overviews some of the algorithms with performance guarantees, as well as some of the algorithms with state-of-the-art performances in practical regimes. Finally, the paper concludes with a few open problems.

show abstract

Section: A Resultssupporting

confidence: 57%

“…IV-A RM codes achieve capacity at low rate [10], [43] RM codes polarize and Twin-RM codes achieve capacity on any BMS [17] . .…”

Section: Capacity Resultsmentioning

confidence: 98%

“…Sberlo and Shpilka [43] polished the techniques of [10] and managed to obtain good estimates for every degree.…”

Section: A Resultsmentioning

confidence: 99%

Section: A Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Reed-Muller Codes: Theory and Algorithms

Abbé

Shpilka

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Finally, we derive an upper bound on the rate of the largest (1, ∞)-RLL subcodes of the specific RM codes that we had used in our lower bounds. Our novel method of analysis uses properties of the weight distribution of RM codes-a topic that has received revived attention over the last decade (see, for example, the survey [12] and the papers [13], [14] and [15]). We hope that our techniques will prove useful in deriving upper bounds for other (𝑑, 𝑘)-and (𝑑, ∞)-RLL constraints, and will be extended, in future work, to any sequence of RM codes that is capacity-achieving over the unconstrained BMS channel.…”

Section: Introductionmentioning

confidence: 99%

On the Performance of Reed-Muller Codes Over $(d,\infty)$-RLL Input-Constrained BMS Channels

Rameshwar¹,

Kashyap²

2022

Preprint

View full text Add to dashboard Cite

This paper considers the input-constrained binary memoryless symmetric (BMS) channel, without feedback. The channel input sequence respects the (𝑑, ∞)-runlength limited (RLL) constraint, which mandates that any pair of successive 1s be separated by at least 𝑑 0s. We consider the problem of designing explicit codes for such channels. In particular, we work with the Reed-Muller (RM) family of codes, which were shown by Reeves and Pfister (2021) to achieve the capacity of any unconstrained BMS channel, under bit-MAP decoding. We show that it is possible to pick (𝑑, ∞)-RLL subcodes of a capacity-achieving (over the unconstrained BMS channel) sequence of RM codes such that the subcodes achieve, under bit-MAP decoding, rates of 𝐶•2 − log 2 (𝑑+1) , where 𝐶 is the capacity of the BMS channel. Finally, we also introduce techniques for upper bounding the rate of any (1, ∞)-RLL subcode of a specific capacity-achieving sequence of RM codes.

show abstract

Reed-Muller Codes Polarize

Abbé

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

View full text Add to dashboard Cite

Reed-Muller (RM) codes and polar codes are generated by the same matrix G m = 1 0 1 1 ⊗m but using different subset of rows. RM codes select simply rows having largest weights. Polar codes select instead rows having the largest conditional mutual information proceeding top to down in G m ; while this is a more elaborate and channel-dependent rule, the top-to-down ordering has the advantage of making the conditional mutual information polarize, giving directly a capacity-achieving code on any binary memoryless symmetric channel (BMSC). RM codes are yet to be proved to have such property.In this paper, we reconnect RM codes to polarization theory. It is shown that proceeding in the RM code ordering, i.e., not top-to-down but from the lightest to the heaviest rows in G m , the conditional mutual information again polarizes. We further demonstrate that it does so faster than for polar codes. This implies that G m contains another code, different than the polar code and called here the twin code, that is provably capacity-achieving on any BMSC. This proves a necessary condition for RM codes to achieve capacity on BMSCs. It further gives a sufficient condition if the rows with largest conditional mutual information correspond to the heaviest rows, i.e., if the twin code is the RM code. We show here that the two codes bare similarity with each other and give further evidence that they are likely the same. E. Abbe is with the Mathematics Institute and the School of Computer and Communication Sciences at EPFL, Switzerland, and the Program in Applied and Computational Mathematics and the Department of Electrical Engineering in Princeton University, USA. M. Ye is with . 1 See [1] for accounts on this conjecture. 2 Recall that a BMS channel is a channel W : {0, 1} → Y such that there is a permutation π on the output alphabet Y satisfying i) π −1 = π and ii) W (y|1) = W (π(y)|0) for all y ∈ Y. arXiv:1901.11533v1 [cs.IT] 31 Jan 2019 : A ⊆ [m]). Next we define another n i.i.d. Bernoulli-1/2 random variables XWe transmit X (m) z , z ∈ {0, 1} m through n independent copies of a BMS channel W : {0, 1} → Y, and we denote the corresponding channel outputs as Y (m,W ) z , z ∈ {0, 1} m . Let X (m) := (X (m) z : z ∈ {0, 1} m ) and Y (m,W ) := (Y (m,W ) z : z ∈ {0, 1} m ).Since W is symmetric and (X (m) z = n(1 − I(W )).

show abstract

On the Performance of Reed-Muller Codes with respect to Random Errors and Erasures

Cited by 4 publications

References 0 publications

Reed-Muller Codes: Theory and Algorithms

Reed-Muller Codes: Theory and Algorithms

On the Performance of Reed-Muller Codes Over $(d,\infty)$-RLL Input-Constrained BMS Channels

Reed-Muller Codes Polarize

Contact Info

Product

Resources

About