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(x_k)). 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^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 [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 probability 1/k^α. Finally we show the applicability of the new tests to PCPs. Starting with a 2-query PCP over an alphabet Σ and with soundness error 1-δ, Rao [19] (building on Raz's (k-fold) parallel repetition theorem [20] and Holenstein's proof [13]) obtains a new 2-query PCP over the alphabet Σ^k with soundness error exp(-δ² k). 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 [8], but with simpler analysis and exponentially better soundness error.