# Randomness-Efficient Low-Degree Tests and Short PCPs Via Epsilon-Biased Sets

 Title: Randomness-Efficient Low-Degree Tests and Short PCPs Via Epsilon-Biased Sets Author: Vadhan, Salil; Ben-Sasson, Eli; Sudan, Madhu; Wigderson, Avi Note: Order does not necessarily reflect citation order of authors. Citation: Ben-Sasson, Eli, Madhu Sudan, Salil Vadhan, and Avi Wigderson. Randomness-efficient low degree tests and short PCPs via epsilon-biased sets. In Proceedings of the 35th Annual ACM Symposium on the Theory of Computing: San Diego, California, USA, June 9-11, 2003, 612-621. New York, NY: ACM Press. Full Text & Related Files: Vadhan_EffShortPCPepsilon.pdf (232.3Kb; PDF) Abstract: We present the first explicit construction of Probabilistically Checkable Proofs (PCPs) and Locally TestableCodes (LTCs) of fixed constant query complexity which have almost-linear (=n^{1+o(1)}) size. Such objects were recently shown to exist (nonconstructively) by Goldreich and Sudan [2002]. The key to these constructions is a nearly optimal randomness-efficient version of the low degree test [Rubinfeld & Sudan 96]. In a similar way we give a randomness-efficient version of the BLR linearity test [Blum, Luby, Rubinfeld 93] (which is used, for instance, in locally testing the Hadamard code). The derandomizations are obtained through \eps-biased sets for vector spaces over finite fields. The analysis of the derandomized tests rely on alternative views of \eps-biased sets --- as generating sets of Cayley expander graphs for the low-degree test, and as defining linear error-correcting codes for the linearity test. Published Version: http://dx.doi.org/10.1145/780542.780631 Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:2961580 Downloads of this work: