We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. ... more >>>

We present a single, common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT) that have been hitherto solved using diverse tools and techniques. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied ... more >>>

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 ecient, but can still be used in many of these applications.
Using the new code we obtain exponential improvements over several ... more >>>