### Revision(s):

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Revision #2 to TR24-036 | 6th April 2024 20:00
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#### A stronger bound for linear 3-LCC

**Abstract:**
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 $n = 2^{\Omega(k^{1/3})}$.

**Changes to previous version:**
- Added analysis for the case of larger alphabets

- Added a proof sketch for Fact 2.3

- Minor changes

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Revision #1 to TR24-036 | 26th February 2024 11:44
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#### A stronger bound for linear 3-LCC

**Abstract:**
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 $n = 2^{\Omega(k^{1/3})}$.

### Paper:

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TR24-036 | 21st February 2024 10:50
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#### A stronger bound for linear 3-LCC

**Abstract:**
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 $n = 2^{\Omega(k^{1/3})}$.