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The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up incurred by this encoding, i.e., how long is the PCP. The state-of-the-art work of Ben-Sasson and Sudan (SICOMP 38(2)) and Dinur (J. ACM 54(3)) shows that for any language in NTIME(t), one can construct PCPs of length $t poly \log t$ that can be verified using a constant number of queries. In this work, we show that if the query complexity is relaxed to polynomial, then one can construct PCPs of length $O(n)$ for circuit-SAT, and PCPs of length $O(t \log t)$ for any language in NTIME(t); both constructions are over a constant-size alphabet.

More specifically, for any $k > 0$ we present (non-uniform) probabilistically checkable proofs (PCPs) for circuit-SAT of length $n2^{O(1/k)}$ that can be checked using $n^k$ queries for circuit-SAT instances of size $n$. Our PCPs have perfect completeness and constant soundness. This is the first constant-rate PCP construction that achieves constant soundness with nontrivial query complexity ($o(n)$).

Our proof replaces the low-degree polynomials in algebraic PCP constructions with tensors of transitive algebraic geometry (AG) codes. We show that the automorphisms of an AG code can be used to simulate the role of affine transformations which are crucial in earlier high-rate algebraic PCP constructions. Using this observation we conclude that any asymptotically good family of transitive AG codes over a constant-sized alphabet leads to a family of constant-rate PCPs with polynomially small query complexity. We construct such codes for every message length, after they have been constructed for infinitely many message lengths by Stichtenoth (Trans. Information Theory 2006).

Joint work with Eli Ben-Sasson, Swastik Kopparty, Or Meir and Henning Stichtenoth.