We give a {\em deterministic} algorithm for approximately computing the fraction of Boolean assignments that satisfy a degree-$2$ polynomial threshold function. Given a degree-2 input polynomial $p(x_1,\dots,x_n)$ and a parameter $\eps > 0$, the algorithm approximates
\[
\Pr_{x \sim \{-1,1\}^n}[p(x) \geq 0]
\]
to within an additive $\pm \eps$ in ... more >>>

Let $g: \{-1,1\}^k \to \{-1,1\}$ be any Boolean function and $q_1,\dots,q_k$ be any degree-2 polynomials over $\{-1,1\}^n.$ We give a \emph{deterministic} algorithm which, given as input explicit descriptions of $g,q_1,\dots,q_k$ and an accuracy parameter $\eps>0$, approximates \[
\Pr_{x \sim \{-1,1\}^n}[g(\sign(q_1(x)),\dots,\sign(q_k(x)))=1] \]
to within an additive $\pm \eps$. For any constant ... more >>>

We construct an explicit family of linear rank-metric codes over any field ${\mathbb F}_h$ that enables efficient list decoding up to a fraction $\rho$ of errors in the rank metric with a rate of $1-\rho-\epsilon$, for any desired $\rho \in (0,1)$ and $\epsilon > 0$. Previously, a Monte Carlo construction ... more >>>