Asymptotically-Good RLCCs with (log n)²⁺o(1) Queries

Recently, Kumar and Mon reached a significant milestone by constructing asymptotically good relaxed locally correctable codes (RLCCs) with poly-logarithmic query complexity. Specifically, they constructed n-bit RLCCs with O(log⁶⁹ n) queries. Their construction relies on a clever reduction to locally testable codes (LTCs), capitalizing on recent breakthrough works in LTCs. As for lower bounds, Gur and Lachish (SICOMP 2021) proved that any asymptotically-good RLCC must make Ω(^{e √}log n) queries. Hence emerges the intriguing question regarding the identity of the least value ¹₂ ≤ e ≤ 69 for which asymptotically-good RLCCs with query complexity (log n)^e+o⁽¹⁾ exist. In this work, we make substantial progress in narrowing the gap by devising asymptotically-good RLCCs with a query complexity of (log n)²⁺o⁽¹⁾. The key insight driving our work lies in recognizing that the strong guarantee of local testability overshoots the requirements for the Kumar-Mon reduction. In particular, we prove that we can replace the LTCs by “vanilla” expander codes which indeed have the necessary property: local testability in the code’s vicinity.