A Stronger Bound for Linear 3-LCC

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