Extractors from Reed-Muller Codes

Finding explicit extractors is an important derandomization goal that has received a lot of attention in the past decade. This research has focused on two approaches, one related to hashing and the other to pseudorandom generators. A third view, regarding extractors as good error correcting codes, was noticed before. Yet, researchers had failed to build extractors directly from a good code, without using other tools from pseudorandomness. We succeed in constructing an extractor directly from a Reed-Muller code. To do this, we develop a novel proof technique. Furthermore, our construction is the first and only construction with degree close to linear. In contrast, the best previous constructions had brought the log of the degree within a constant of optimal, which gives polynomial degree. This improvement is important for certain applications. For example, it follows that approximating the VC dimension to within a factor of N1 δ is AM-hard for any positive δ. Some of this work was done while the author was at the University of California at Berkeley, and supported in part by a David and Lucile Packard Fellowship for Science and Engineering and NSF NYI Grant CCR-9457799. †Some of this work was done while the author was on leave at the University of California at Berkeley. Supported in part by a David and Lucile Packard Fellowship for Science and Engineering, NSF Grant CCR-9912428, NSF NYI Grant CCR-9457799, and an Alfred P. Sloan Research Fellowship.