Moshkovitz, Dana; Raz, Ran Sub-constant error probabilistically checkable proof of almost-linear size. (English) Zbl 1213.68317 Comput. Complexity 19, No. 3, 367-422 (2010). Summary: We show a construction of a \(PCP\) with both sub-constant error and almost-linear size. Specifically, for some constant \(0 < \alpha < 1\), we construct a \(PCP\) verifier for checking satisfiability of Boolean formulas that on input of size \(n\) uses \(\log n+O((\log n)^{1-\alpha})\) random bits to make 7 queries to a proof of size \(n\cdot 2^{O((\log n)^{1-\alpha})}\), where each query is answered by \(O((\log n)^{1-\alpha})\) bit long string, and the verifier has perfect completeness and error \(2^{-\Omega((\log n)^{\alpha})}\). The construction is by a new randomness-efficient version of the aggregation through curves technique. Its main ingredients are a recent low degree test with both sub-constant error and almost-linear size and a new method for constructing a short list of balanced curves. Cited in 5 Documents MSC: 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 68Q60 Specification and verification (program logics, model checking, etc.) Keywords:probabilistically checkable proof (PCP); almost-linear size; sub-constant error; balanced curves PDFBibTeX XMLCite \textit{D. Moshkovitz} and \textit{R. Raz}, Comput. Complexity 19, No. 3, 367--422 (2010; Zbl 1213.68317) Full Text: DOI