Revised: January 1, 2021
Published: June 6, 2022
Abstract: [Plain Text Version]
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 direct proof that improves some of the parameters of the main result of Guo, Haramaty, and Sudan (FOCS 2015), showing robust soundness of lifted codes.
To obtain the above transformation, we relate the notions of local testing and robust testing to the notion of agreement testing that attempts to find out whether valid partial assignments can be stitched together to a global codeword. We first show that agreement testing implies robust testing, and then show that local testing implies agreement testing. Our proof is combinatorial, and is based on sampling properties of the collection of local views of local testers. Thus, it immediately applies to local testers of lifted codes that query random affine subspaces in $\F_q^m$, and moreover seems amenable to extension to other families of locally testable codes with expanding families of local views.
A conference version of this paper appeared in the Proceedings of the 10th Innovations in Theoretical Computer Science Conference, 2019.