We construct the first asymptotically good relaxed locally correctable codes with polylogarithmic query complexity, bringing the upper bound polynomially close to the lower bound of Gur and Lachish (SICOMP 2021). Our result follows from showing that a high-rate locally testable code can boost the block length of a smaller relaxed locally correctable code, while preserving the correcting radius and incurring only a modest additive cost in rate and query complexity. We use the locally testable code's tester to check if the amount of corruption in the input is low; if so, we can "zoom-in" to a suitable substring of the input and recurse on the smaller code's local corrector. Hence, iterating this operation with a suitable family of locally testable codes due to Dinur, Evra, Livne, Lubotzky, and Mozes (STOC 2022) yields asymptotically good codes with relaxed local correctability, arbitrarily large block length, and polylogarithmic query complexity.

Our codes asymptotically inherit the rate and distance of any locally testable code used in the final invocation of the operation. Therefore, our framework also yields nonexplicit relaxed locally correctable codes with polylogarithmic query complexity that have rate and distance approaching the Gilbert-Varshamov bound.

Improved exposition, changed notation

We cement the intuitive connection between relaxed local correctability and local testing by presenting a concrete framework for building a relaxed locally correctable code from any family of linear locally testable codes with sufficiently high rate. When instantiated using the locally testable codes of Dinur et al. (STOC 2022), this framework yields the first asymptotically good relaxed locally correctable and decodable codes with polylogarithmic query complexity, which finally closes the superpolynomial gap between query lower and upper bounds. Our construction combines high-rate locally testable codes of various sizes to produce a code that is locally testable at every scale: we can gradually "zoom in" to any desired codeword index, and a local tester at each step certifies that the next, smaller restriction of the input has low error.

Our codes asymptotically inherit the rate and distance of any locally testable code used in the final step of the construction. Therefore, our technique also yields nonexplicit relaxed locally correctable codes with polylogarithmic query complexity that have rate and distance approaching the Gilbert-Varshamov bound.