Nearly Tight Lower Bounds for Relaxed Locally Decodable Codes via Robust Daisies

Abstract: 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 $n = k^{1+Ω(1/q)}$. This bound closely matches the known upper bound of $n = k^{1+O(1/q)}$ by Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (STOC 2004). Our proof introduces the notion of robust daisies, which are relaxed sunflowers with pseudorandom structure, and leverages a new spread lemma to extract dense robust daisies from arbitrary distributions.