Unbalanced Expanders and Randomness Extractors from Parvaresh-Vardy Codes

We give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree (which is polylogarithmic in the number of vertices). Both the degree and the number of right-hand vertices are polynomially close to optimal, whereas the previous constructions of Ta-Shma, Umans, and Zuckerman (STOC ‘01) required at least one of these to be quasipolynomial in the optimal. Our expanders have a short and self-contained description and analysis, based on the ideas underlying the recent list-decodable error-correcting codes of Parvaresh and Vardy (FOCS ‘05). Our expanders can be interpreted as near-optimal “randomness condensers,” that reduce the task of extracting randomness from sources of arbitrary min-entropy rate to extracting randomness from sources of min-entropy rate arbitrarily close to 1, which is a much easier task. Using this connection, we obtain a new, self-contained construction of randomness extractors that is opt...

• Type: Conference
• Year: 2007
• Where: COCO
• Authors: Venkatesan Guruswami, Christopher Umans, Salil P. Vadhan
• Keywords: COCO 2007, Improved Explicit Construction, Randomness, Unbalanced Bipartite Expander