From Local to Robust Testing via Agreement Testing

Abstract

A local tester for an error-correcting code is a probabilistic procedure that queries a small (sublinear) subset of coordinates, accepts codewords with probability one, and rejects non-codewords with probability proportional to their distance from the code. The local tester is said to be robust if for non-codewords it satisfies the stronger condition that the average distance of local views from accepting views is proportional to the distance from the code. Robust testing is an important component in constructions of locally testable codes and probabilistically checkable proofs as it allows for composition of local tests. We show that for certain codes, any (natural) local tester can be converted to a robust tester with roughly the same number of queries. Our result holds for the class of affine-invariant lifted codes which is a broad class of codes that includes Reed–Muller codes, as well as recent constructions of high-rate locally testable codes (Guo, Kopparty, and Sudan, ITCS 2013). Instantiating this with known local testing results for lifted codes gives a more.