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 probabilistically checkable proofs (PCPs). Goldreich and Safra [SIAM J. Comput., 29(2000), pp. 1132-1154] pioneered its local testing and its PCP application. A recent result by Dinur and Goldenberg [Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 613-622] 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^α 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 ^α). Our techniques (based on recent simplified decoding algorithms for the same code [R. Impagliazzo et al., Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, 2008, pp. 579-588]) also allow us to considerably simplify the analysis of the 2-query test of [Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 613-622]. We then show how to derandomize their test, achieving a code of polynomial rate, independent of k, and success probability 1/k ^α. Finally, we show the applicability of the new tests to PCPs. Starting with a 2-query PCP with a projection property over an alphabet Σ and with soundness error 1 - δ, Rao [Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, 2008, pp. 1-10] (building on Raz's (k-fold) parallel repetition theorem [R. Raz, SIAM J. Comput., 27(1998), pp. 763-803] and Holenstein's proof [T. Holenstein, Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, 2007, pp. 411-419] obtains a new 2-query PCP over the alphabet Σ^k with soundness error exp(-δ²k). Our techniques yield a 2-query PCP with soundness error exp(-δ√k). Our PCP construction turns out to be essentially the same as the miss-match proof system defined and analyzed by Feige and Kilian [SIAM J. Comput., 30(2000), pp. 324-346] but with simpler analysis and exponentially better soundness error.