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

In 1948, Shannon used a probabilistic argument to show that there exist codes achieving a maximal rate defined by the channel capacity. In 1954, Muller and Reed introduced a simple deterministic code construction, conjectured shortly after to achieve channel capacity. Major progress was made towards establishing this conjecture over the last decades, with various branches of discrete mathematics involved such as combinatorial bounds, sharp thresholds, hypercontractivity, additive combinatorics and polarization theory. In particular, the special case of the erasure channel was settled by Kudekar at al., relying on Bourgain-Kalai’s sharp threshold theorem for symmetric monotone properties. The main case of error channels remained however unsettled, due in particular to the property being non-monotone and the lack of techniques to obtain fast local error decay up to capacity, despite the notable vanishing bound on the local error from Reeves-Pfister. This paper closes the conjecture’s proof. The main ingredient is a new recursive boosting framework for coding, where codewords are decoded by aggregating restrictions on a ‘subspace-sunflower’ structure, analogous to the structure from Erdős-Rado 1960. The dependencies between the sunflower petals are handled with an $L_{2}$ and $L_{4}$ Boolean Fourier analysis, and a list-decoding argument with a weight enumerator bound from Sberlo-Shpilka is used to control the global error from the local one. For the local error, while monotonicity does not apply, we show that a ‘weak threshold’ result still holds using solely symmetries. This gives in particular a shortened and tightened argument for the vanishing local error result, and with prior works, it also implies the strong wire-tap secrecy of RM codes on pure-state classical-quantum channels.