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 show a nearly optimal lower bound on the length of linear relaxed locally decodable codes (RLDCs). Specifically, we prove that any $q$-query linear RLDC $C\colon \{0,1\}^k \to \{0,1\}^n$ must satisfy $n = k^{1+\Omega(1/q)}$. This bound closely matches the known upper bound of $n = k^{1+O(1/q)}$ by Ben-Sasson, Goldreich, ... more >>>