Making the Long Code Shorter

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The long code is a central tool in hardness of approximation, especially in questions related to the Unique Games Conjecture. We construct a new code that is exponentially more efficient, but can still be used in many of these applications. Using the new code we obtain exponential improvements over several known results, including the following: (1) For any $\varepsilon>0$, we show the existence of an $n$-vertex graph $G$ where every set of $o(n)$ vertices has expansion $1-\varepsilon$, but $G$'s adjacency matrix has more than $\exp(\log^{\delta}n)$ eigenvalues larger than $1-\varepsilon$, where $\delta$ depends only on $\varepsilon$. This answers an open question of Arora, Barak, and Steurer [Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, 2010, pp. 563--572], who asked whether one can improve over the noise graph on the Boolean hypercube that has ${\rm poly}(\log n)$ such eigenvalues. (2) A gadget that reduces Unique Games instances with linear constraints modulo $K$ into instances with alphabet $k$ with a blowup of $k^{{\rm polylog}(K)}$, improving over the previously known gadget with blowup of $k^{\Omega(K)}$. (3) An $n$-variable integrality gap for Unique Games that survives $\exp({\rm poly}(\log\log n))$ rounds of the semidefinite programming version of the Sherali--Adams hierarchy, improving on the previously known bound of ${\rm poly}(\log\log n)$. We show a connection between the local testability of linear codes and Small-Set Expansion in certain related Cayley graphs and use this connection to derandomize the noise graph on the Boolean hypercube.