We prove that for every odd $q\geq 3$, any $q$-query binary, possibly non-linear locally decodable code ($q$-LDC) $E:\{\pm 1\}^k \rightarrow \{\pm 1\}^n$ must satisfy $k \leq \tilde{O}(n^{1-2/q})$. For even $q$, this bound was established in a sequence of works (Katz and Trevisan (2000), Goldreich, Karloff, Schulman, and Trevisan (2002), and ... more >>>

We prove that level-$\ell$ Kikuchi graphs of random $2r$-uniform hypergraphs spectrally approximate Kikuchi graph of the complete $2r$-uniform hypergraph at a sampling rate that is sharp up to a logarithmic factor, in the regime $r\leq \ell \leq n/2$. Our proof is based on the matrix Bernstein inequality, but, unlike prior ... more >>>

We prove that any linear $3$-query locally correctable code $C \colon \mathbb{F}_2^k \to \mathbb{F}_2^n$ with constant correction radius must have length $n \geq 2^{\Omega(\sqrt{k})}$. This improves on the prior best lower bound of $2^{\Omega(\sqrt{k/\log k})}$, due to [AG24], and shows that the folklore construction of linear $3$-LCCs from Reed--Muller codes, ... more >>>