New direct-product testers and 2-query PCPs

The "direct product code" of a function f gives its values on all k-tuples (f(x1), . . . , f(xk)). This basic construct underlies "hardness amplification" in cryptography, circuit complexity and PCPs. Goldreich and Safra [12] pioneered its local testing and its PCP application. A recent result by Dinur and Goldenberg [5] enabled for the first time testing proximity to this important code in the "list-decoding" regime. In particular, they give a 2-query test which works for polynomially small success probability 1/k , and show that no such test works below success probability 1/k. Our main result is a 3-query test which works for exponentially small success probability exp(-k ). Our techniques (based on recent simplified decoding algorithms for the same code [15]) also allow us to considerably simplify the analysis of the 2-query test of [5]. We then show how to derandomize their test, achieving a code of polynomial rate, independent of k, and success prob...

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: Russell Impagliazzo, Valentine Kabanets, Avi Wigderson
• Keywords: Algorithms, Better Soundness Error, Small Success Probability, Soundness Error, STOC 2009