  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].
