Robust PCPs of proximity, shorter PCPs and applications to coding

We continue the study of the trade-o between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satisability of circuits of size n): 1. We present PCPs of length exp(o(log log n) 2)n that can be veried by making o(log log n) Boolean queries. 2. For every "> 0, we present PCPs of length exp(log n) n that can be veried by making a constant number of Boolean queries. In both cases, false assertions are rejected with constant probability (which may be set to be arbitrarily close to 1). The multiplicative overhead on the length of the proof, introduced by transforming a proof into a probabilistically checkable one, is just quasi-polylogarithmic in the rst case (of query complexity o(log log n)), and 2 (log n), for any "> 0, in the second case (of constant query complexity). Our techniques include the introduction of a new variant of PCPs that we call \Robust PCPs of proximity". These new PCPs facilitate proof composition, which is a central ingredient in construction of PCP systems. (A related notion and its composition properties were discovered independently by Dinur and Reingold.) Our main technical contribution is a construction of a \length-ecient " Robust PCP of proximity. While the new construction uses many of the standard techniques in PCPs, it does dier from previous constructions in fundamental ways, and in particular does not use the \parallelization" step of Arora et al. The alternative approach may be of independent interest. We also obtain analogous quantitative results for locally testable codes. In addition, we introduce a relaxed notion of locally decodable codes, and present such codes mapping k information bits to codewords of length