cryptography protocol designers

Abstract: We prove that the covering radius of the third-order Reed--Muller code in eight variables is fifty-six. The proof separates even- and odd-weight inputs by moment-controlled slicing. On the even side, a transport invariant on quadratic-label stars and consistency of singleton restrictions give a compatible cubic correction. On the odd side, symmetry of the third moment yields a covariance-rank obstruction. Its rank-two exception is a quartic plus three points of a two-plane; the zero-second-moment case is a prescribed-point perturbation of a quintic. We prove that every such point perturbation has third-order nonlinearity at most fifty-five by a marked derivative identity and complete restriction-code covers. Consequently the maximum among exact-degree-eight functions is fifty-five. The structural reductions are proved analytically, and complete finite certificates verify the explicitly stated local covering and compatibility assertions. The matching lower bound follows from the published quartic relative radius.

Abstract: For a balanced Boolean function $f$ on the discrete hypercube, we study the $α$-stability $\mathbb{E}[(T_ρf)^α]$, where $T_ρ$ is the Boolean noise operator and $0<α<1$. Using the asymmetric Hellinger kernel estimates recently established by OpenAI, we prove that dictators minimize the $7/8$-stability in every dimension and at every correlation $0\leρ\le1$. At interior correlations they are the only minimizers. As a consequence, dictators minimize the $α$-stability for every $7/8\leα<1$, maximize the $α$-stability for every $1<α<2$, and maximize the one-sided Shannon entropy stability $\mathbb{E}[T_ρf\log T_ρf]$. This resolves a conjecture of Li and Médard and strengthens the Courtade--Kumar conjecture in the balanced setting.