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.

翻译：我们构造了首批具有多对数查询复杂度的渐近优松弛局部可纠错码，将上界多项式地逼近于Gur与Lachish（SICOMP 2021）提出的下界。该结果基于以下发现：高码率的局部可测试码能够提升较小松弛局部可纠错码的码块长度，同时保持纠错半径并仅引入码率与查询复杂度上的适度附加开销。我们利用局部可测试码的测试器检查输入中的污染量是否较低；若是，则可"缩放"至输入的合适子串，并对较小码的局部纠错器进行递归操作。因此，通过迭代运用Dinur、Evra、Livne、Lubotzky与Mozes（STOC 2022）提出的特定局部可测试码族，我们获得了具有松弛局部可纠错性、任意长码块长度及多对数查询复杂度的渐近优码。我们的码渐近地继承了最终操作中使用的任意局部可测试码的码率与距离。因此，本框架还生成了非显式的松弛局部可纠错码，其查询复杂度为多对数，且码率与距离逼近吉尔伯特-瓦尔沙莫夫界。