Randomness-efficient low degree tests and short PCPs via epsilon-biased sets

We present the first explicit construction of Probabilistically Checkable Proofs (PCPs) and Locally Testable Codes (LTCs) of fixed constant query complexity which have almost-linear (= n · 2^{Õ(√log n)} size. Such objects were recently shown to exist (nonconstructively) by Goldreich and Sudan. Previous explicit constructions required size n^1+ω(ε) with 1/ε queries. The key to these constructions is a nearly optimal randomness-efficient version of the low degree test. In a similar way we give a randomness-efficient version of the BLR linearity test (which is used, for instance, in locally testing the Hadamard code). The derandomizations are obtained through ε-biased sets for vector spaces over finite fields. The analysis of the derandomized tests rely on alternative views of ε-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.