## Abstract

The PCP theorem [Arora et al. 1998; Arora and Safra 1998] 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, that is, how long is the PCP compared to the original NP-proof? The state-of-the-art work of Ben-Sasson and Sudan [2008] and Dinur [2007] shows that one can encode proofs of length n by PCPs of length n·poly log n that can be verified using a constant number of queries. In this work, we show that if the query complexity is relaxed to n^{ε}, 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). More specifically, for any ε > 0, we present (nonuniform) probabilistically checkable proofs (PCPs) of length 2^{O(1/ε)}·n that can be checked using n^{ε} 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 that 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. Such codes are constructed in the appendix to this article for the first time for every message length, building on an earlier construction for infinitely many message lengths by Stichtenoth [2006].

Original language | English |
---|---|

Article number | 32 |

Journal | Journal of the ACM |

Volume | 63 |

Issue number | 4 |

DOIs | |

State | Published - Nov 2016 |

### Bibliographical note

Funding Information:Eli Ben-Sasson's research received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement number 240258, the US-Israel Binational Science Foundation (grant 2014359), and the Israeli Science Foundation (grant 1501/14). Swastik Kopparty's research supported by the National Science Foundation under grants NSF CCF-1253886 and NSF CCF-1540634, the US-Israel Binational Science Foundation under grant 2014359, and a Sloan Fellowship. Or Meir's research supported by the National Science Foundation under grant CCF-0832797. Henning Stichtenoth's research supported by Tubitak, proj. no. 111T234. This research was done while Eli Ben-Sasson was a visitor at MIT, and Or Meir was a post-doctoral fellow at the Institute for Advanced Study. This work was partially done at the Oberwolfach Workshop on Complexity Theory, 2012. We are grateful to the organizers for the chance to attend the workshop, and to the center for the wonderful hospitality. We thank the anonymous referees for valuable comments that helped clarify the exposition.

Publisher Copyright:

© 2016 ACM.

## Keywords

- Algebraic-geometric codes
- Error-correcting codes
- PCPs
- Probabilistic proof systems
- Sublinear time algorithms

## ASJC Scopus subject areas

- Software
- Control and Systems Engineering
- Information Systems
- Hardware and Architecture
- Artificial Intelligence