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].