This work proves new results on the ability of binary Reed-Muller codes to decode from random errors and erasures. We obtain these results by proving improved bounds on the weight distribution of Reed-Muller codes of high degrees.Specifically, given weight β ∈ (0, 1) we prove an upper bound on the number of codewords of relative weight at most β. We obtain new results in two different settings: for weights β < 1/2 and for weights that are close to 1/2. Our results for weights close to 1/2 also answer an open problem posed by Beame et al. [BGY18].Our new bounds on the weight distribution imply that RM codes with m variables and degree γm, for some explicit constant γ, achieve capacity for random erasures (i.e. for the binary erasure channel) and for random errors (for the binary symmetric channel). Earlier, it was known that RM codes achieve capacity for the binary symmetric channel for degrees r = o(m). For the binary erasure channel it was known that RM codes achieve capacity forThus, our result provide a new range of parameters for which RM achieve capacity for these two well studied channels. In addition, our results imply that for every ǫ > 0 (in fact we can get up to ǫ = Ω √ log m √ m ) RM codes of degree r < (1/2 − ǫ)m can correct a fraction of 1 − o(1) random erasures with high probability. We also show that, information theoretically, such codes can handle a fraction of 1 2 − o(1) random errors with high probability. Thus, for example, given noisy evaluations of a degree 0.499m polynomial, it is possible to interpolate it even if a random 0.499 fraction of the evaluations were corrupted, with high probability. While the o(1) terms are not the correct ones to ensure capacity, these results show that RM codes of such degrees are in some sense close to achieving capacity. A Combinatorial lemmas 41 B Proof of main lemmas of [KLP12] 42 C Missing calculations from the proof of Theorem 1.4 44 1 Fq denotes the field with q elements 2 We only consider RM codes over F2 as this is the most difficult and interesting case for the questions we study. Pr [A 2 ] = ν≥(1−ǫ)(1−p)2 m Pr [|S| = ν2 m ] · Pr B| S = ν2 m . Clearly, the probability Pr B | |S| = ν2 m gets larger as ν gets smaller hence, Pr [A 2 ] = ν≥(1−ǫ)(1−p)2 mPr S |S| = ν2 m · Pr B||S| = ν2 m ≤ ν≥(1−ǫ)(1−p)2 m Pr |S| = ν2 m · Pr B | |S| = (1 − p)(1 − ǫ)2 m ≤ Pr S B | |S| = (1 − p)(1 − ǫ)2 m .

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Using techniques and results from [8] we strengthen the bounds of [10] on the weight distribution of linear codes achieving capacity on the BEC. In particular, we show that for any doubly transitive binary linear code C ⊆ {0, 1} n of rate 0 < R < 1 with weight distribution (a 0 , ..., a n ) holdsFor doubly transitive codes with minimal distance at least Ω (n c ), 0 < c ≤ 1, the error factor of 2 o(n) in this bound can be removed at the cost of replacing 1 − R with a smaller constant a = a(R, c) < 1 − R. Moreover, in the special case of Reed-Muller codes, due to the additional symmetries of these codes, this error factor can be removed at essentially no cost.This implies that for any doubly transitive code C of rate R with minimal distance at least Ω (n c ), there exists a positive constant p = p(R, c) such that C decodes errors on BSC(p) with high probability if p < p(R, c). For doubly transitive codes of a sufficiently low rate (smaller than some absolute constant) the requirement on the minimal distance can be omitted, and hence this critical probability p(R) depends only on R. Furthermore, p(R) → 1 2 as R → 0. In particular, a Reed-Muller code C of rate R decodes errors on BSC(p) with high probability if1 4 ln 2 , answering a question posed in [1].

Reed-Muller codes were introduced in 1954, with a simple explicit construction based on polynomial evaluations, and have long been conjectured to achieve Shannon capacity on symmetric channels. Major progress was made towards a proof over the last decades; using combinatorial weight enumerator bounds, a breakthrough on the erasure channel from sharp thresholds, hypercontractivity arguments, and polarization theory. Another major progress recently established that the bit error probability vanishes slowly below capacity. However, when channels allow for errors, the results of Bourgain-Kalai do not apply for converting a vanishing bit to a vanishing block error probability, neither do the known weight enumerator bounds. The conjecture that RM codes achieve Shannon capacity on symmetric channels, with high probability of recovering the codewords, has thus remained open.This paper closes the conjecture's proof. It uses a new recursive boosting framework, which aggregates the decoding of codeword restrictions on 'subspace-sunflowers', handling their dependencies via an L p Boolean Fourier analysis, and using a list-decoding argument with a weight enumerator bound from Sberlo-Shpilka. The proof does not require a vanishing bit error probability for the base case, but only a non-trivial probability, obtained here for general symmetric codes. This gives in particular a shortened and tightened argument for the vanishing bit error probability result of Reeves-Pfister, and with prior works, it implies the strong wire-tap secrecy of RM codes on pure-state classical-quantum channels.