Extractors: Optimal up to constant factors

This paper provides the first explicit construction of extractors which are simultaneously optimal up to constant factors in both seed length and output length. More precisely, for every n, k, our extractor uses a random seed of length O(log n) to transform any random source on n bits with (min-)entropy k, into a distribution on (1- α)k bits that is ε-close to uniform. Here α and ε can be taken to be any positive constants. (In fact, ε can be almost polynomially small). Our improvements are obtained via three new techniques, each of which may be of independent interest. The first is a general construction of mergers from locally decodable error-correcting codes. The second introduces new condensers that have constant seed length (and retain a constant fraction of the min-entropy in the random source). The third is a way to augment the "win-win repeated condensing" paradigm of with error reduction techniques like so that the our constant seed-length condensers can be used without error accumulation.