This paper strengthens the low-error PCP characterization of NP, coming closer to the upper limit of the BGLR conjecture. Consider the task of verifying a witness for the membership of a given input in an NP language, using a constant number of accesses. We show that it is possible to achieve an error probability exponentially small in the number of bits accessed, where the number of bits in each access is as high as log β n , for any constant β < 1. The BGLR conjecture asserts the same for a

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... nstant β where β ≤ 1. Our results are in fact stronger, implying that the Gap-Quadratic-Solvability problem with a constant number of variables in each equation is NP-hard. That is, given a system of n quadratic-equations over a field F of size up to 2 log β n , where each equation depends on a constant number of variables, it is NP-hard to distinguish between the case where there is a common solution to all of the equations, and the case where any assignment satisfies at most a 2 |F | fraction of them. At the same time, our proof presents a direct construction of a low-degree-test whose error-probability is exponentially small in the