Abstract:
We consider the problem of randomness extraction from independent sources. We construct an extractor that can extract from a constant number of independent sources of length $n$, each of which have min-entropy $n^\gamma$ for an arbitrarily small constant $\gamma > 0$. Our result is different from recent work for this problem in the sense that it does not rely on any results from additive number theory. Our extractor is built by composing previous constructions of strong seeded extractors in simple ways. Using Bourgain's extractor as a black box, we obtain a new extractor for 2 independent blockwise sources with few blocks, even when the min-entropy is as small as $\polylog(n)$. We also show how to modify the 2 source disperser for linear min-entropy of Barak et al. and the 3 source extractor of Raz to get dispersers/extractors with exponentially small error and linear output length where previously both were constant. In terms of Ramsey Hypergraphs, for every constant $1> \gamma >0$ our construction gives a family of explicit $O(1/\gamma)$-uniform hypergraphs on $N$ vertices that avoid cliques and independent sets of size $2^{(\log N)^\gamma}$.

Abstract:
We consider the problem of bit extraction from independent sources. We construct an extractor that can extract from a constant number of independent sources of length $n$, each of which have min-entropy $n^\gamma$ for an arbitrarily small constant $\gamma > 0$. Our constructions are different from recent extractor constructions \cite{BarakIW04, BarakKSSW05, Raz05, Bourgain05} for this problem in the sense that they do not rely on any results from additive number theory. They are obtained by composing previous constructions of strong seeded extractors in simple ways.