In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist binary LCCs and LTCs with block length $n$, constant rate (which can even be taken arbitrarily close to 1), constant ... more >>>

For a boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$, let $\hat{f}$ be the unique multilinear polynomial such that $f(x)=\hat{f}(x)$ holds for every $x\in \{0,1\}^n$. We show that, assuming $\hbox{VP}\not=\hbox{VNP}$, there exists a polynomial-time computable $f$ such that $\hat{f}$ requires super-polynomial arithmetic circuits. In fact, this $f$ can be taken as a monotone 2-CNF, ... more >>>

We present a complete classification of all possible sets of classical reversible gates acting on bits, in terms of which reversible transformations they generate, assuming swaps and ancilla bits are available for free. Our classification can be seen as the reversible-computing analogue of Post's lattice, a central result in mathematical ... more >>>