Affine extractors over prime fields

Abstract

An affine extractor is a map that is balanced on every affine subspace of large enough dimension. We construct an explicit affine extractor AE from $\mathbb{F}^n$ to $\mathbb{F}$, $\mathbb{F}$ a prime field, so that AE(x) is exponentially close to uniform when x is chosen uniformly at random from an arbitrary affine subspace of $\mathbb{F}^n$ of dimension at least δn, 0<δ≤1 a constant. Previously, Bourgain constructed such affine extractors when the size of $\mathbb{F}$ is two. Our construction is in the spirit of but different than Bourgain’s construction. This allows for simpler analysis and better quantitative results.