A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP

We introduce a new low-degree--test, one that uses the restriction of low-degree polynomials to planes (i. e. , affine sub-spaces of dimension 2), rather than the restriction to lines (i. e. , affine sub-spaces of dimension 1). We prove the new test to be of a very small errorprobability (in particular, much smaller than constant). The new test enables us to prove a low-error characterization of NP in terms of PCP. Specifically, our theorem states that, for any given ffl? 0, membership in any NP language can be verified with O (1) accesses, each reading logarithmic number of bits, and such that the error-probability is 2 \ log 1\ n. Our results are in fact stronger, as stated below. One application of the new characterization of NP is that approximating SET-COVER to within a logarithmic factors is NP-hard. Previous analysis for low-degree-tests, as well as previous characterizations of NP in terms of PCP, have managed to achieve, with constant number of accesses, error. . .