Locally testable codes (LTCs) are error-correcting codesthat admit very efficient codeword tests. An LTC is said tobe strong if it has a proximity-oblivious tester;that is, a tester that makes only a constant number of queriesand reject non-codewords with probability that depends solelyon their distance from the code.

Locally decodable codes (LDCs) are complimentary to LTCs.While LTCs allow for highly efficient rejection ofstrings that are far from being codewords, LDCs allow for highlyefficient recovery of individual bits of the information thatis encoded in strings that are close to being codewords.

While there are known constructions of strong-LTCswith nearly-linear length, the existence of a constant-query LDCwith polynomial length is a major open problem.In an attempt to bypass this barrier, Ben-Sasson et al (SICOMP 2006)introduced a natural relaxation of local decodability,called relaxed-LDCs. This notion requires local recoveryof nearly all individual information-bits,yet allows for recovery-failure (but not error) on the rest.Ben-Sasson et al constructed a constant-query relaxed-LDCwith nearly-linear length (i.e., length k1+for an arbitrarily small constant 0">0,where k is the dimension of the code).

This work focuses on obtaining strong testability and relaxeddecodability simultaneously. We construct a family of binary linearcodes of nearly-linear length that are both strong-LTCs(with one-sided error) and constant-query relaxed-LDCs.This improves upon the previously known constructions,which obtain either weak LTCs or require polynomial length.

Our construction heavily relies on tensor codes and PCPs.In particular, we provide strong canonical PCPs of proximityfor membership in any linear code with constant rate and relative distance.Loosely speaking, these are PCPs of proximity wherein the verifieris proximity oblivious (similarly to strong-LTCs)and every valid statement has a unique canonical proof.Furthermore, the verifier is required to reject non-canonical proofs(even for valid statements).

As an application, we improve the best known separation resultbetween the complexity of decision and verification in the settingof property testing