We give explicit constructions of extractors which work for a source of
any min-entropy on strings of length n. These extractors can extract any
constant fraction of the min-entropy using O(log^2 n) additional random
bits, and can extract all the min-entropy using O(log^3 n) additional
random bits. Both of these ... more >>>

Weak designs were defined by Raz, Reingold and Vadhan (1999) and are
used in constructions of extractors. Roughly speaking, a weak design
is a collection of subsets satisfying some near-disjointness
properties. Constructions of weak designs with certain parameters are
given in [RRV99]. These constructions are explicit in the sense that
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A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been introduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a $t-(n,k,?)$ combinatorial design with tiny, yet ... more >>>

The $\delta$-Coin Problem is the computational problem of distinguishing between coins that are heads with probability $(1+\delta)/2$ or $(1-\delta)/2,$ where $\delta$ is a parameter that is going to $0$. We study the complexity of this problem in the model of constant-depth Boolean circuits and prove the following results.

1. Upper ... more >>>