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(log^{69} 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 Ω̃(√{log n}) queries. Hence emerges the intriguing question regarding the identity of the least value 1/2 ≤ 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.

A q-locally correctable code (LCC)$C:\{0,1\}^{k}\rightarrow \{0,1\}^{n}$is a code in which it is possible to correct every bit of a (not too) corrupted codeword by making at most$q$queries to the word. The cases in which$q$is constant are of special interest, and so are the cases that$C$is linear. In a breakthrough result Kothari and Manohar (STOC 2024) showed that for linear 3-LCC$n=2^{\Omega(k^{1/8})}$. In this work we prove that$n=2^{\Omega(k^{1/4})}$. As Reed-Muller codes yield 3-LCC with$n=2^{O(k^{1/2})}$, this brings us closer to closing the gap. Moreover, in the special case of design-LCC (into which Reed-Muller fall) the bound we get is$n=2^{\Omega(k^{1/3})}$.

In their highly influential paper, Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (STOC 2004) introduced the notion of a relaxed locally decodable code (RLDC). Similarly to a locally decodable code (Katz-Trevisan; STOC 2000), the former admits access to any desired message symbol with only a few queries to a possibly corrupted codeword. An RLDC, however, is allowed to abort when identifying corruption. The natural analog to locally correctable codes, dubbed relaxed locally correctable codes (RLCC), was introduced by Gur, Ramnarayan and Rothblum (ITCS 2018) who constructed asymptotically-good length-nRLCC and RLDC with $(\log n)^{O(\log\log n)}$ queries.In this work we construct asymptotically-good RLDC and RLCC with an improved query complexity of $(\log n)^{O(\log\log\log n)}$. To achieve this, we devise a mechanism-an alternative to the tensor product-that squares the length of a given code. Compared to the tensor product that was used by Gur et al. and by many other constructions, our mechanism is significantly more efficient in terms of rate deterioration, allowing us to obtain our improved construction.

A locally correctable code (LCC) is an error correcting code that allows correction of any arbitrary coordinate of a corrupted codeword by querying only a few coordinates. We show that any {\em zero-error} $2$-query locally correctable code $\mathcal{C}: \{0,1\}^k \to Σ^n$ that can correct a constant fraction of corrupted symbols must have $n \geq \exp(k/\log|Σ|)$. We say that an LCC is zero-error if there exists a non-adaptive corrector algorithm that succeeds with probability $1$ when the input is an uncorrupted codeword. All known constructions of LCCs are zero-error. Our result is tight upto constant factors in the exponent. The only previous lower bound on the length of 2-query LCCs over large alphabet was $Ω\left((k/\log|Σ|)^2\right)$ due to Katz and Trevisan (STOC 2000). Our bound implies that zero-error LCCs cannot yield $2$-server private information retrieval (PIR) schemes with sub-polynomial communication. Since there exists a $2$-server PIR scheme with sub-polynomial communication (STOC 2015) based on a zero-error $2$-query locally decodable code (LDC), we also obtain a separation between LDCs and LCCs over large alphabet. For our proof of the result, we need a new decomposition lemma for directed graphs that may be of independent interest. Given a dense directed graph $G$, our decomposition uses the directed version of Szemerédi regularity lemma due to Alon and Shapira (STOC 2003) to partition almost all of $G$ into a constant number of subgraphs which are either edge-expanding or empty.

Since they were first introduced by Schulman (STOC 1993), the construction of tree codes remained an elusive open problem. The state-of-the-art construction by Cohen, Haeupler and Schulman (STOC 2018) has constant distance and (logn)e colors for some constant e > 1 that depends on the distance, where n is the depth of the tree. Insisting on a constant number of colors at the expense of having vanishing distance, Gelles, Haeupler, Kol, Ron-Zewi, and Wigderson (SODA 2016) constructed a distance Ω(1/logn) tree code.

The main contribution of this work is a rate amplification procedure for LCC. Our procedure converts any q-query linear LCC, having rate ρ and, say, constant distance to an asymptotically good LCC with q^poly(1/ρ) queries. Our second contribution is a distance amplification procedure for LDC that converts any linear LDC with distance δ and, say, constant rate to an asymptotically good LDC. The query complexity only suffers a multiplicative overhead that is roughly equal to the query complexity of a length 1/δ asymptotically good LDC. This improves upon the poly(1/δ) overhead obtained by the AEL distance amplification procedure [Alon and Luby, 1996; Alon et al., 1995]. Our work establishes that the construction of asymptotically good LDC and LCC is reduced, with a minor overhead in query complexity, to the problem of constructing a vanishing rate linear LCC and a (rapidly) vanishing distance linear LDC, respectively.