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In 1948, Shannon used a probabilistic argument to prove the existence of codes achieving channel capacity. In 1954, Muller and Reed introduced a simple deterministicÌýcodeÌýconstruction, conjectured shortly after to achieve channel capacity. Major progress was made towards establishing this conjecture over the last decades, with various areasÌýof discreteÌýmathematics involved. In particular, the special case of the erasure channel was settled in 2015 by Kudekar at al., relying on Bourgain-Kalai's sharp threshold theoremÌýfor symmetricÌýmonotone properties. The main case of error channels remained however open, due in particular to the property being non-monotone, and the absence of techniques toÌýobtain fast local errorÌýdecay. In this talk, we provide a proof of the general conjecture. The proof circumvents the use of Bourgain-Kalai's theorem by establishing first a "weak"Ìýthreshold property that relies solelyÌýon symmetries. The proof then proceeds with a boosting framework to improve the weak local error into a strong global error, relying on sunflowerÌýstructures asÌýdefined inÌýErdÅ‘s-Rado but for subspaces. Joint work withÌýColin Sandon.