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Extractors from Reed-Muller codes. (English) Zbl 1094.68036
Summary: Finding explicit extractors is an important derandomization goal that has received a lot of attention in the past decade. Previous 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 to achieve degree close to linear. In contrast, the best previous constructions 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 was used [E. Mossel and C. Umans, “On the complexity of approximating the VC dimension”, J. Comput. Syst. Sci. 65, 660–671 (2002; Zbl 1059.68049)] to show that approximating VC dimension to within a factor of \(N^{1 - \delta }\) is AM-hard for any positive \(\delta\) .

MSC:
94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
68W99 Algorithms in computer science
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