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

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.