Relaxed locally correctable codes with improved parameters

Locally decodable codes (LDCs) are error-correcting codes $C : \Sigma^k \to \Sigma^n$ that admit a local decoding algorithm that recovers each individual bit of the message by querying only a few bits from a noisy codeword. An important question in this line of research is to understand the optimal trade-off between the query complexity of LDCs and their block length. Despite importance of these objects, the best known constructions of constant query LDCs have super-polynomial length, and there is a significant gap between the best constructions and the known lower bounds in terms of the block length. For many applications it suffices to consider the weaker notion of relaxed LDCs (RLDCs), which allows the local decoding algorithm to abort if by querying a few bits it detects that the input is not a codeword. This relaxation turned out to allow decoding algorithms with constant query complexity for codes with almost linear length. Specifically, [Ben-Sasson et al. (2006)] constructed a $q$-query RLDC that encodes a message of length $k$ using a codeword of block length $n = O_q(k^{1+O(1/\sqrt{q})})$ for any sufficiently large $q$, where $O_q(\cdot)$ hides some constant that depends only on $q$. In this work we improve the parameters of [Ben-Sasson et al. (2006)] by constructing a $q$-query RLDC that encodes a message of length $k$ using a codeword of block length $O_q(k^{1+O(1/q)})$ for any sufficiently large $q$. This construction matches (up to a multiplicative constant factor) the lower bounds of [Katz and Trevisan (2000), Woodruff (2007)] for constant query LDCs, thus making progress toward understanding the gap between LDCs and RLDCs in the constant query regime. In fact, our construction extends to the stronger notion of relaxed locally correctable codes (RLCCs), introduced in [Gur et al. (2018)], where given a noisy codeword the correcting algorithm either recovers each individual bit of the codeword by only reading a small part of the input, or aborts if the input is detected to be corrupt.