Storing Information With Extractors

We deal with the problem of storing a set of K elements that are taken from a large universe of size N, such that membership in the set can be determined with high probability by looking at just one bit of the representation. Buhrman et al. show an explicit construction with about K-2 log N storing bits. We show an explicit construction with about K1+o(1) storing bits, that gets closer to the optimal K log N bound.Our technique is of independent interest. Buhrman et al. show a non-explicit optimal (up to constant factors) construction that is based on the existence of certain good unbalanced expanders. To make the construction explicit one needs to be able to explicitly 'encode' and 'decode' such expanding graphs. We generalize the notion of loss-less condensers of Ta-Shma et al. [Proc. 33rd Annual ACM Symposium on Theory of Computing, 2001, pp. 143-152] and build such graphs. We further show how to efficiently decode such graphs using an observation from Ta-Shma and Zuckerman [Manuscript, 2001] about Trevisan's extractor. (C) 2002 Elsevier Science B.V. All rights reserved.