A code $C \colon \{0,1\}^k \to \{0,1\}^n$ is a $q$-locally decodable code ($q$-LDC) if one can recover any chosen bit $b_i$ of the message $b \in \{0,1\}^k$ with good confidence by randomly querying the encoding $x := C(b)$ on at most $q$ coordinates. Existing constructions of $2$-LDCs achieve $n = \exp(O(k))$, and lower bounds show that this is in fact tight. However, when $q = 3$, far less is known: the best constructions achieve $n = \exp(k^{o(1)})$, while the best known results only show a quadratic lower bound $n \geq \tilde{\Omega}(k^2)$ on the blocklength. In this paper, we prove a near-cubic lower bound of $n \geq \tilde{\Omega}(k^3)$ on the blocklength of $3$-query LDCs. This improves on the best known prior works by a polynomial factor in $k$. Our proof relies on a new connection between LDCs and refuting constraint satisfaction problems with limited randomness. Our quantitative improvement builds on the new techniques for refuting semirandom instances of CSPs developed in [GKM22, HKM23] and, in particular, relies on bounding the spectral norm of appropriate Kikuchi matrices.

翻译：编码 $C \colon \{0,1\}^k \to \{0,1\}^n$ 称为 $q$ 元局部可解码编码（$q$-LDC），若可通过至多 $q$ 个坐标随机查询编码 $x:=C(b)$，以高置信度恢复消息 $b \in \{0,1\}^k$ 的任意选定比特 $b_i$。现有 $2$-LDC 构造达到 $n = \exp(O(k))$，且下界表明此结果紧致。然而当 $q=3$ 时，已知结果显著不足：最佳构造实现 $n = \exp(k^{o(1)})$，而现有最优下界仅给出二次下界 $n \geq \tilde{\Omega}(k^2)$。本文证明三元查询 LDC 的码长满足近立方下界 $n \geq \tilde{\Omega}(k^3)$，将已知最优结果的多项式因子改进至 $k$ 的三次方。证明依赖于 LDC 与有限随机性约束满足问题（CSP）反驳之间的新联系，其定量改进基于 [GKM22, HKM23] 中发展的半随机 CSP 实例反驳技术，并特别依赖于对 Kikuchi 矩阵谱范数的界。