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.

翻译：我们通过提出一个具体框架，从任意具有足够高码率的线性局部可检验码族构造松弛局部可校正码，从而加固了松弛局部可校正性与局部检验之间的直观联系。当使用Dinur等人（STOC 2022）的局部可检验码实例化时，该框架首次生成具有多对数查询复杂度的渐近最优松弛局部可校正与可译码，最终弥合了查询下界与上界之间的超多项式差距。我们的构造通过组合不同规模的高码率局部可检验码，生成在每一尺度上均可进行局部检验的码：可逐步"放大"到任意所需码字索引，每一步的局部检验器均验证输入的下一个较小限制具有低错误率。我们的码渐近继承构造最后一步所用局部可检验码的码率与最小距离。因此，该技术还生成了具有多对数查询复杂度、码率与最小距离趋近吉尔伯特-瓦尔沙莫夫界的非显式松弛局部可校正码。