We give two new characterizations of ($\F_2$-linear) locally testable error-correcting codes in terms of Cayley graphs over $\F_2^h$:

\begin{enumerate}
\item A locally testable code is equivalent to a Cayley graph over $\F_2^h$ whose set of generators is significantly larger than $h$ and has no short linear dependencies, but yields a ... more >>>

A theorem of Håstad shows that for every constraint satisfaction problem (CSP) over a fixed size domain, instances where each variable appears in at most $O(1)$ constraints admit a non-trivial approximation algorithm, in the sense that one can beat (by an additive constant) the approximation ratio achieved by the naive ... more >>>

We study the approximability of two natural Boolean constraint satisfaction problems: Horn satisfiability and exact hitting set. Under the Unique Games conjecture, we prove the following optimal inapproximability and approximability results for finding an assignment satisfying as many constraints as possible given a {\em
near-satisfiable} instance.

\begin{enumerate}
\item ... more >>>

We study lower bounds for Locality Sensitive Hashing (LSH) in the strongest setting: point sets in $\{0,1\}^d$ under the Hamming distance. Recall that $\mathcal{H}$ is said to be an $(r, cr, p, q)$-sensitive hash family if all pairs $x,y \in \{0,1\}^d$ with dist$(x,y) \leq r$ have probability at least $p$ ... more >>>