New direct-product testers and 2-query PCPs

Abstract

The "direct product code" of a function f gives its values on all k-tuples (f (x 1 ), . . . , f (x k )). This basic construct underlies "hardness amplification" in cryptography, circuit complexity and PCPs. Goldreich and Safra [GS00] pioneered its local testing and its PCP application. A recent result by Dinur and Goldenberg [DG08] 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 [IJKW08]) also allow us to considerably simplify the analysis of the 2-query test of [DG08]. 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 [Rao08] (building on Raz's (k-fold) parallel repetition theorem [Raz98] and Holenstein's proof [Hol07]) obtains a new 2-query PCP over the alphabet Σ k with soundness error exp(−δ 2 k). Our techniques yield a 2-query PCP with soundness error exp(−δ √ k); this is incomparable to Rao's result [Rao08] since our bound also applies when k < 1/δ. Our PCP construction turns out to be essentially the same as the miss-match proof system defined and analyzed by Feige and Kilian [FK00], but with simpler analysis and exponentially better soundness error. *

Composition of Low-Error 2-Query PCPs Using Decodable PCPs

“…al. [IKW09] obtained a related derandomization. While their derandomization result applies only to direct products and not to the construction of PCPs, this direction seems promising.…”

Towards lower bounds on locally testable codes via density arguments

“…Roughly speaking, LTCs are invariably constructed by starting with a decent "base-construction" of an LTC (such as a Hadamard, Reed-Muller, or constant-blocklength code) and modifying it by various techniques like repetition (Raz 1998), concatenation (Arora et al 1998;Arora & Safra 1998), tensoring (Ben-Sasson & Sudan 2008), gap-amplification (Dinur 2007), taking direct-products (Dinur & Goldenberg 2008;Impagliazzo et al 2009) and PCPP-composition (Ben-Sasson et al 2006;Dinur & Reingold 2006). These operations improve the LTC-related parameters of the code, and they increase soundness and/or reduce query complexity but none of them increases rate.…”