We give improved lower bounds for binary $3$-query locally correctable codes (3-LCCs) $C \colon \{0,1\}^k \rightarrow \{0,1\}^n$. Specifically, we prove: (1) If $C$ is a linear design 3-LCC, then $n \geq 2^{(1 - o(1))\sqrt{k} }$. A design 3-LCC has the additional property that the correcting sets for every codeword bit form a perfect matching and every pair of codeword bits is queried an equal number of times across all matchings. Our bound is tight up to a factor $\sqrt{8}$ in the exponent of $2$, as the best construction of binary $3$-LCCs (obtained by taking Reed-Muller codes on $\mathbb{F}_4$ and applying a natural projection map) is a design $3$-LCC with $n \leq 2^{\sqrt{8 k}}$. Up to a $\sqrt{8}$ factor, this resolves the Hamada conjecture on the maximum $\mathbb{F}_2$-codimension of a $4$-design. (2) If $C$ is a smooth, non-linear $3$-LCC with near-perfect completeness, then, $n \geq k^{\Omega(\log k)}$. (3) If $C$ is a smooth, non-linear $3$-LCC with completeness $1 - \varepsilon$, then $n \geq \tilde{\Omega}(k^{\frac{1}{2\varepsilon}})$. In particular, when $\varepsilon$ is a small constant, this implies a lower bound for general non-linear LCCs that beats the prior best $n \geq \tilde{\Omega}(k^3)$ lower bound of [AGKM23] by a polynomial factor. Our design LCC lower bound is obtained via a fine-grained analysis of the Kikuchi matrix method applied to a variant of the matrix used in [KM23]. Our lower bounds for non-linear codes are obtained by designing a from-scratch reduction from nonlinear $3$-LCCs to a system of "chain polynomial equations": polynomial equations with similar structure to the long chain derivations that arise in the lower bounds for linear $3$-LCCs [KM23].

翻译：我们改进了二进制3-查询局部可纠错码（3-LCCs）$C \colon \{0,1\}^k \rightarrow \{0,1\}^n$ 的下界。具体而言，我们证明：(1) 若 $C$ 是线性设计3-LCC，则 $n \geq 2^{(1 - o(1))\sqrt{k}}$。设计3-LCC具有附加性质：每个码字比特的纠错集构成完美匹配，且所有匹配中任意两个码字比特的被查询次数相等。该下界在2的指数上紧至因子$\sqrt{8}$，因为最优二进制3-LCC构造（通过对$\mathbb{F}_4$上的Reed-Muller码施加自然投影映射得到）是满足$n \leq 2^{\sqrt{8 k}}$的设计3-LCC。除去$\sqrt{8}$因子，这解决了关于4-设计最大$\mathbb{F}_2$-余维数的Hamada猜想。(2) 若 $C$ 是光滑非线性3-LCC且具有近完美完备性，则 $n \geq k^{\Omega(\log k)}$。(3) 若 $C$ 是光滑非线性3-LCC且完备性为$1 - \varepsilon$，则 $n \geq \tilde{\Omega}(k^{\frac{1}{2\varepsilon}})$。特别地，当$\varepsilon$为小常数时，该下界将一般非线性LCC的下界多项式地优于先前的最佳结果 $n \geq \tilde{\Omega}(k^3)$（[AGKM23]）。我们的设计LCC下界通过对[KM23]中矩阵变体应用Kikuchi矩阵方法的精细分析获得。非线性码的下界则通过设计从非线性3-LCC到"链多项式方程组"的全新归约获得：该方程组的结构与线性3-LCC下界[KM23]中出现的长链推导类似。