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Revision #1 to TR05-106 | 5th December 2005 00:00
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#### Extractors for a Constant Number of Polynomial Small Min-Entropy Independent Sources

**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 $2^{(\log N)^\gamma}$.

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TR05-106 | 26th September 2005 00:00
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#### Extractors for a Constant Number of Polynomial Min-Entropy Independent Sources

TR05-106
Authors:

Anup Rao
Publication: 28th September 2005 15:13

Downloads: 2581

Keywords:

**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.