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Randomness-efficient low-degree tests and short PCPs

via epsilon-biased sets

Eli Ben-Sasson, Madhu Sudan, Salil Vadhan, and Avi Wigderson

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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.

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