Abstract
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 · 2Õ(√log n) size. Such objects were recently shown to exist (nonconstructively) by Goldreich and Sudan. Previous explicit constructions required size n1+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.
Original language | English |
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Pages (from-to) | 612-621 |
Number of pages | 10 |
Journal | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |
DOIs | |
State | Published - 2003 |
Event | 35th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States Duration: 9 Jun 2003 → 11 Jun 2003 |
Keywords
- Linearity Testing
- Locally Testable Codes
- Low Degree Testing
- Probabilistically Checkable Proofs
- Property Testing