A proof that Reed-Muller codes achieve Shannon capacity on symmetric channels

Abbe, Emmanuel; Sandon, Colin

doi:10.1109/focs57990.2023.00020

Cited by 7 publications

(5 citation statements)

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“…Using estimates for the magnitude of Krawtchouk polynomials and bounds for the weight distribution of the dual code C ⊥ , one can thus bound the norm ∥f ∥ 2 2 in the set-up where the error string z is a random vector of weight exactly ϵN . Using essentially the same techniques, one can also bound the norm ∥f ∥ 2 2 when the error string z is a random vector of weight ≈ ϵN , i.e. z is taken uniformly at random from the set S = {x ∈ F N 2 : wt(x) = ϵN ± N 3/4 }.…”

Section: List Decoding Resultsmentioning

confidence: 99%

“…For this reason, when Arikan showed that polar codes achieve capacity over the BSC, Reed-Muller codes received renewed attention from the coding theory community. A long and fruitful line of work [4,38,6,25,1,47,50] has recently culminated in Abbe and Sandon showing that Reed-Muller codes achieve capacity over all BMS channels [2].…”

Section: List Decoding Resultsmentioning

confidence: 99%

“…An important open question is thus whether general doubly transitive codes achieve capacity over all BMS channels under block-MAP decoding, or whether one really needs the additional symmetry that Reed-Muller codes possess. Some of the key techniques used in [47] and [2] are very much tailored to Reed-Muller codes, or at least to codes consisting of evaluations of polynomials over F N 2 ; in order to prove the same results for arbitrary doubly transitive codes, it may be necessary to develop a more general framework.…”

Section: List Decoding Resultsmentioning

confidence: 99%

“…Using fundamental results from Fourier analysis about the influences of symmetric boolean functions [29,55,14] has led to a very successful line of work, with [38] showing that Reed-Muller codes achieve capacity over the BEC and [47,2] showing that they achieve capacity over the BSC. In fact, [38] show that for any s > 1, if a linear code C ⊆ F N 2 of size |C| ≤ 2 (1−ϵ)N has a doubly-transitive symmetry group G such that for every S ⊆ {1, 2, .…”

mentioning

confidence: 99%

“…In some regimes, it improves on all previously known weight bounds for Reed-Muller codes (See Table 1 and Appendix A for a comparison of our weight bound with previous results). 2. We give a new criterion to validate that a code can be decoded over the BSC.…”

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confidence: 99%

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A criterion for decoding on the binary symmetric channel

Rao,

Sprumont

2025

AMC

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We present an approach to showing that a linear code is resilient to random errors. We use this approach to obtain decoding results for both transitive and doubly transitive codes. We give three kinds of results about linear codes in general, and transitive linear codes in particular.1. We give a tight bound on the weight distribution of every transitive linear code2. We give a criterion that certifies that a linear code C can be decoded on the binary symmetric channel. Let Ks(x) denote the Krawtchouk polynomial of degree s, and let C ⊥ denote the dual code of C. We show that bounds on E c∈C ⊥ [K ϵN (wt(c)) 2 ] imply that C recovers from errors on the binary symmetric channel with parameter ϵ. Weaker bounds can be used to obtain list-decoding results using similar methods. One consequence of our criterion is that whenever the weight distribution of C ⊥ is sufficiently close to the binomial distribution in some interval around N 2 , C is resilient to ϵ-errors. 3. We combine known estimates for the Krawtchouk polynomials with our weight bound for transitive codes, and with known weight bounds for doubly transitive codes, to obtain list-decoding results for both these families of codes. In some regimes, our bounds for doubly transitive codes achieve the information-theoretic optimal trade-off between rate and list size.

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Section: List Decoding Resultsmentioning

confidence: 99%