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

2025-11-26

AI Summary

This work establishes a tight lower bound on the length of linear relaxed locally decodable codes (RLDCs). For $q$-query linear RLDCs $C:{0,1}^k o {0,1}^n$, we prove the first lower bound $n = k^{1+Omega(1/q)}$, nearly matching the best-known upper bound $k^{1+O(1/q)}$ and yielding an essentially tight characterization. Our method introduces the “robust daisy” — a novel combinatorial structure combining pseudorandomness with high-density local dependencies — and couples it with a new diffusion lemma that efficiently extracts dense local dependency patterns from arbitrary query distributions. Integrating tools from combinatorial design, probabilistic analysis, and coding theory, our approach breaks prior reliance on algebraic structures. This yields the first near-optimal, general-purpose lower bound on the complexity of linear RLDCs, resolving a central open problem in the study of relaxed locality.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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 $Ccolon {0,1}^k o {0,1}^n$ 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.

Problem

• Lower bound on linear relaxed locally decodable codes length
• Matching known upper bounds for RLDCs via robust daisies
• Extracting dense robust daisies using a spread lemma

Innovation

• Introduces robust daisies with pseudorandom structure
• Leverages a new spread lemma for extraction
• Proves nearly optimal lower bounds for linear RLDCs