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

TL;DR

This work proves a nearly tight lower bound for linear q-query RLDCs, showing $n rac{k}{ ext{log}^2 k} n$ bound of the form $n = k^{1+Ω(1/q)}$, which matches the BGHSV upper bound up to constants. The authors introduce robust daisies, a pseudorandom structure with a kernel outside which the set-system is spread, and prove a Robust Daisy Lemma to extract dense robust daisies from arbitrary distributions via a Spreadness Extraction Lemma. They define $(m,k)$-spreadness, prove a Small-Set Spread Lemma, and develop a global-sampler framework that leverages robust daisies to compress information and derive lower bounds. The results extend to linear RLDCs via Goldberg’s reduction and highlight a fundamental $k^{Θ(1/q)}$ spreadness barrier, suggesting the core difficulty lies in pseudorandom structure outside a small kernel. Overall, the paper advances lower-bound techniques for RLDCs and introduces novel combinatorial tools with potential relevance to sunflower-type phenomena and spreadness in probabilistic combinatorics.

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.

Theorems & Definitions (50)

• Theorem 1
• Corollary 2
• Corollary 3
• Definition 1.1
• Lemma 1.2
• Lemma 2.1
• Definition 2.2: $(m, k)$-spread distributions
• Lemma 2.3: "The Small-Set Spread Lemma"; informally stated, see lemma:spread_distribution
• Lemma 2.4
• Lemma 3.2: Janson's Inequality AS16