We prove that the blocklength $n$ of a linear $3$-query locally correctable code (LCC) $\mathcal{L} \colon {\mathbb F}^k \to {\mathbb F}^n$ with distance $\delta$ must be at least $n \geq 2^{\Omega\left(\left(\frac{\delta^2 k}{(|{\mathbb F}|-1)^2}\right)^{1/8}\right)}$. In particular, the blocklength of a linear $3$-query LCC with constant distance over any small field grows exponentially with $k$. This improves on the best prior lower bound of $n \geq \tilde{\Omega}(k^3)$ [AGKM23], which holds even for the weaker setting of $3$-query locally decodable codes (LDCs), and comes close to matching the best-known construction of $3$-query LCCs based on binary Reed-Muller codes, which achieve $n \leq 2^{O(k^{1/2})}$. Because there is a $3$-query LDC with a strictly subexponential blocklength [Yek08, Efr09], as a corollary we obtain the first strong separation between $q$-query LCCs and LDCs for any constant $q \geq 3$. Our proof is based on a new upgrade of the method of spectral refutations via Kikuchi matrices developed in recent works [GKM22, HKM23, AGKM23] that reduces establishing (non-)existence of combinatorial objects to proving unsatisfiability of associated XOR instances. Our key conceptual idea is to apply this method with XOR instances obtained via long-chain derivations, a structured variant of low-width resolution for XOR formulas from proof complexity [Gri01, Sch08].

翻译：我们证明，对于线性三查询局部可纠码（LCC）$\mathcal{L} \colon {\mathbb F}^k \to {\mathbb F}^n$，若其距离为$\delta$，则码长$n$必须满足$n \geq 2^{\Omega\left(\left(\frac{\delta^2 k}{(|{\mathbb F}|-1)^2}\right)^{1/8}\right)}$。特别地，在任意小域上具有恒定距离的线性三查询LCC的码长随$k$呈指数增长。这改进了此前最优下界$n \geq \tilde{\Omega}(k^3)$ [AGKM23]（该下界甚至适用于更弱的局部可解码码（LDC）设定），并逼近基于二元Reed-Muller码的最优三查询LCC构造（其码长为$n \leq 2^{O(k^{1/2})}$）。由于存在严格次指数码长的三查询LDC [Yek08, Efr09]，作为推论，我们首次证明了对于任意常数$q \geq 3$，$q$查询LCC与LDC之间存在强分离。我们的证明基于对近期工作[GKM22, HKM23, AGKM23]中开发的Kikuchi矩阵谱反驳方法的新升级——该方法将组合对象的存在性判定归约为相关XOR实例的可满足性否定。关键技术突破在于：采用通过长链推导（证明复杂性领域XOR公式低宽度解析结构变体[Gri01, Sch08]）获得的XOR实例来应用此方法。