A Lower Bound for Relaxed Locally Decodable Codes

TL;DR

This paper establishes a fundamental lower bound on the blocklength of relaxed locally decodable codes, showing they cannot be nearly linear in message length, thus resolving a long-standing open problem.

Contribution

It proves a lower bound that rules out nearly linear blocklengths for O(1)-query relaxed LDCs, advancing understanding of their theoretical limits.

Findings

Proves O(1)-query relaxed LDCs cannot have blocklength n = k^{1+ o(1)}

Resolves an open problem posed by Goldreich in 2004

Establishes fundamental limits on the efficiency of relaxed LDCs

Abstract

A locally decodable code (LDC) C:{0,1}^k -> {0,1}^n is an error correcting code wherein individual bits of the message can be recovered by only querying a few bits of a noisy codeword. LDCs found a myriad of applications both in theory and in practice, ranging from probabilistically checkable proofs to distributed storage. However, despite nearly two decades of extensive study, the best known constructions of O(1)-query LDCs have super-polynomial blocklength. The notion of relaxed LDCs is a natural relaxation of LDCs, which aims to bypass the foregoing barrier by requiring local decoding of nearly all individual message bits, yet allowing decoding failure (but not error) on the rest. State of the art constructions of O(1)-query relaxed LDCs achieve blocklength n = O(k^{1+ \gamma}) for an arbitrarily small constant \gamma. We prove a lower bound which shows that O(1)-query relaxed LDCs cannot achieve blocklength n = k^{1+ o(1)}. This resolves an open problem raised by Goldreich in 2004.