On the Power of Relaxed Local Decoding Algorithms

Abstract

A locally decodable code (LDC) C: {0, 1}k → {0, 1}n is an error correcting code that admits algorithms for recovering individual bits of the message 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 LDCs with O(1)-query decoding algorithms 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 (k1+γ) for an arbitrarily small constant γ. Using algorithmic and combinatorial techniques, we prove an impossibility result, showing that codes with blocklength n = k1+o(1) cannot be relaxed decoded with O(1)-query algorithms. This resolves an open problem raised by Goldreich in 2004.