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

TL;DR

This paper establishes a nearly tight lower bound on the length of linear relaxed locally decodable codes, matching known upper bounds, by introducing robust daisies and a new spread lemma to analyze code structure.

Contribution

It introduces the concept of robust daisies and a spread lemma to derive nearly optimal lower bounds for linear RLDCs, advancing understanding of code length constraints.

Findings

Lower bound of n = k^{1+Ω(1/q)} for linear RLDCs

Introduction of robust daisies as a new analytical tool

A new spread lemma to extract dense structures from distributions

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 $n = k^{1+\Omega(1/q)}$. 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.