The concept of subspace designs was introduced by Guruswami and Xing (STOC'13), and explicit constructions were given by Guruswami and Kopparty (FOCS'13). These are families of subspaces that have small intersection with any given subspace of a fixed dimension. We introduce \emph{robust subspace designs}. Informally, these are a quantitative extension ... more >>>

We prove that the scaling window of random $k$-SAT has width $O(n^{1/2+1/k})$, a polynomial improvement over our previous bound of $O(n/\log n)$. Combined with a result of Abbe and Montanari, this establishes the satisfiability conjecture for every fixed $k\geq 3$.

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We study cryptographic primitives that are both locally computable (i.e., in $\mathrm{NC}^0$) and exponentially secure. For pseudorandom generators (PRGs) and universal one-way hash functions (UOWHFs), we further require linear stretch and linear compression, respectively, which is essentially the best one can hope for in this setting. Such primitives simultaneously achieve ... more >>>