Reed–Muller capacity conjecture for binary-input memoryless symmetric channels

Let $\mathcal{P}$ be a binary-input memoryless symmetric (BMS) channel with output alphabet $\mathcal{Y}$, and let

$$C(\mathcal{P})=I(U,\mathcal{P})=\frac{1}{2}\sum_{x\in\{0,1\},\,y\in\mathcal{Y}}\mathcal{P}(y\mid x)\log_2\frac{\mathcal{P}(y\mid x)}{\mathcal{P}(y\mid 0)/2+\mathcal{P}(y\mid 1)/2}$$

be its capacity under the uniform input distribution. For a sequence of Reed–Muller codes $\operatorname{RM}(m_i,r_i)$ with rates $R_i={m_i\choose\leq r_i}2^{-m_i}$ tending to a rate $R<C(\mathcal{P})$, the Reed–Muller capacity conjecture. the codewords can be decoded successfully with high probability: any of the $\lfloor 2^{nR_i}\rfloor$ codewords can be decoded with probability $1-o_{m_i}(1)$ despite corruption by $\mathcal{P}$. For the binary symmetric channel $\operatorname{BSC}(\epsilon)$, the capacity is $1-H(\epsilon)$. This conjecture asserts that the explicit Reed–Muller construction achieves Shannon capacity under maximum-likelihood or an appropriate successful decoding procedure; the paper states that it proves the conjecture for all BMS channels, so the claim is solved.