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, 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.
Categories: cs.CC cs.IT math.CO math.IT