A $q$-locally correctable code (LCC) $C:\{0,1\}^k \to \{0,1\}^n$ 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 $n = 2^{\Omega(k^{1/3})}$.

- Added analysis for the case of larger alphabets
- Added a proof sketch for Fact 2.3
- Minor changes

A $q$-locally correctable code (LCC) $C:\{0,1\}^k \to \{0,1\}^n$ 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 $n = 2^{\Omega(k^{1/3})}$.

A $q$-locally correctable code (LCC) $C:\{0,1\}^k \to \{0,1\}^n$ 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 $n = 2^{\Omega(k^{1/3})}$.