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 >>>

Given a circuit $G: \{0, 1\}^n \to \{0, 1\}^m$ with $m > n$, the *range avoidance* problem ($\text{Avoid}$) asks to output a string $y\in \{0, 1\}^m$ that is not in the range of $G$. Besides its profound connection to circuit complexity and explicit construction problems, this problem is also related ... more >>>

Razborov and Rudich's natural proofs barrier roughly says that it is computationally hard to certify that a uniformly random truth table has high circuit complexity. In this work, we show that the natural proofs barrier (specifically, Rudich's conjecture that there are no NP-constructive natural properties against $P/poly$) implies the following ... more >>>