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

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

While there are known constructions of strong-LTCs
with nearly-linear length, the existence of a constant-query LDC
with 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 recovery
of 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-LDC
with nearly-linear length (i.e., length $k^{1 + \alpha}$
for an arbitrarily small constant $\alpha>0$,
where $k$ is the dimension of the code).

This work focuses on obtaining strong testability and relaxed
decodability simultaneously. We construct a family of binary linear
codes 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 proximity
for membership in any linear code with constant rate and relative distance.
Loosely speaking, these are PCPs of proximity wherein the verifier
is 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 result
between the complexity of decision and verification in the setting
of property testing.