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 optimal up to constant
factors, while being much simpler than the previous construction of Lu
et al. (STOC `03) and improving upon it when the error parameter is
small (e.g. 1/poly(n)).
We give new constructions of randomness extractors and lossless condensers that are optimal to within constant factors in both the seed length and the output length. For extractors, this matches the parameters of the current best known construction [LRVW03]; for lossless condensers, the previous best constructions achieved optimality to within a constant factor in one parameter only at the expense of a polynomial loss in the other.
Our constructions are based on the Parvaresh-Vardy codes [PV05], and our proof technique is inspired by the list-decoding algorithm for those codes. The main object we construct is a condenser that loses only the entropy of its seed plus one bit, while condensing to entropy rate $1 - \alpha$ for any desired constant $\alpha > 0$. This construction is simple to describe, and has a short and completely self-contained analysis. Our other results only require, in addition, standard uses of randomness-efficient hash functions (to obtain a lossless condenser) or expander walks (to obtain an extractor).
Our techniques also show for the first time that a natural construction based on univariate polynomials (i.e., Reed-Solomon codes) yields a condenser that retains a $1 - \alpha$ fraction of the source min-entropy, for any desired constant $\alpha > 0$, while condensing to constant entropy rate and using a seed length that is optimal to within constant factors.
