A Near-Cubic Lower Bound for 3-Query Locally Decodable Codes from Semirandom CSP Refutation

A code $C\text{\hspace{0.17em}⁣}:\{0,1{\}}^k\to \{0,1{\}}^n$ 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 $n\ge \tilde{\mathrm{\Omega }}(k^2)$ on the blocklength. In this paper, we prove a near-cubic lower bound of $n\ge \tilde{\mathrm{\Omega }}(k^3)$ 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.