Improved Average-Case Lower Bounds for DeMorgan Formula Size

We give an explicit function h: {0, 1}ⁿ → {0, 1} such that every deMorgan formula of size n^3-o(1)/r² agrees with h on at most a fraction of 1/2+2^-Ω(r) of the inputs. This improves the previous average-case lower bound of Komargodski and Raz (STOC, 2013). Our technical contributions include a theorem that shows that the "expected shrinkage" result of Haastad (SIAM J. Comput., 1998) actually holds with very high probability (where the restrictions are chosen from a certain distribution that takes into account the structure of the formula), combining ideas of both Impagliazzo, Meka and Zuckerman (FOCS, 2012) and Komargodski and Raz. In addition, using a bit-fixing extractor in the construction of h allows us to simplify a major part of the analysis of Komargodski and Raz¹.