# Increasing the Output Length of Zero-Error Dispersers

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## Summary

Let \(\cal C\) be a class of probability distributions over a finite set *Ω*. A function \(D:\Omega \mapsto {\{0,1\}}^m\) is a *disperser* for \(\cal C\) with *entropy threshold* *k* and *error* *ε* if for any distribution *X* in \(\cal C\) such that *X* gives positive probability to at least \(2^k\) elements we have that the distribution *D(X)* gives positive probability to at least \((1-\epsilon)2^m\) elements. A long line of research is devoted to giving explicit (that is, polynomial-time computable) dispersers (and related objects called “extractors”) for various classes of distributions while trying to maximize *m* as a function of *k*.

In this chapter we are interested in explicitly constructing *zero-error dispersers* (that is, dispersers with error \(\epsilon=0\)). For several interesting classes of distributions there are explicit constructions in the literature of zero-error dispersers with “small” output length *m*, and we give improved constructions that achieve “large” output length, namely \(m=\Omega(k)\).

We achieve this by developing a general technique to improve the output length of zero-error dispersers (namely, to transform a disperser with short output length into one with large output length). This strategy works for several classes of sources and is inspired by the transformation that improves the output length of extractors used in previous chapters. However, we stress that this technique is different, and in particular gives nontrivial results in the errorless case.

Using our approach we construct improved zero-error disper- sers for the class of *2-sources*. More precisely, we show that for any constant \(\delta>0\) there is a constant \(\eta>0\) such that for sufficiently large *n* there is a poly-time computable function \(D:{\{0,1\}}^n \times {\{0,1\}}^n \mapsto {\{0,1\}}^{\eta n}\) such that for any two independent distributions \(X_1,X_2\) over \({\{0,1\}}^n\) such that both of them support at least \(2^{\delta n}\) elements we get that the output distribution \(D(X_1,X_2)\) has full support. This improves the output length of previous constructions by [4] and has applications in Ramsey Theory and in constructing certain data structures [24].

We also use our techniques to give explicit constructions of zero-error dispersers for bit-fixing sources and affine sources over polynomially large fields. These constructions improve the best known explicit constructions due to [52, 25] and achieve \(m=\Omega(k)\) for bit-fixing sources and \(m=k-o(k)\) for affine sources.

This chapter is based on [27]

## Keywords

Convex Combination Explicit Construction Good Index Seed Length Good Seed## References

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