Relaxed Locally Correctable Codes with Nearly-Linear Block Length and Constant Query Complexity

Locally correctable codes (LCCs) are error correcting which admit local algorithms that correct any individual symbol of a corrupted codeword via minuscule number queries. For systematic codes, this notion is stronger than locally decodable (LDCs), where the goal to only recover symbols message. One central problems in algorithmic coding theory construct -query LCCs and LDCs with minimal block length. Alas, state-of-the-art such requires super-polynomial length for correction decoding, despite much attention during last two decades. The study relaxed LDCs, allow algorithm abort (but not err) on small fraction locations, provides way circumvent barrier. This relaxation turned out constant-query decoding polynomial Focusing correction, Gur, Ramnarayan, Rothblum [Proceedings 9th Innovations Theoretical Computer Science Conference, ITCS’18, 2018, pp. 1–27] showed there exist achieve nearly-quartic , an arbitrarily constant . We LCC nearly-linear significantly narrows gap between lower bound states no In particular, our construction matches parameters achieved by Ben-Sasson et al. [SIAM J. Comput., 36 (2006), 889–974], who constructed same parameters. resolves open problem raised 1–27].