One of the most important open problems in the theory
of error-correcting codes is to determine the
tradeoff between the rate $R$ and minimum distance $\delta$ of a binary
code. The best known tradeoff is the Gilbert-Varshamov bound,
and says that for every $\delta \in (0, 1/2)$, there are codes
with minimum distance $\delta$ and rate $R = R_{GV}(\delta) > 0$ (for
a certain simple function $R_{GV}(\cdot)$). In this paper
we show that the Gilbert-Varshamov bound can be achieved
by codes which support local error-detection and
error-correction algorithms.

Specifically, we show the following results.

1. Local Testing: For all $\delta \in (0,1/2)$ and all $R < R_{GV}(\delta)$,
there exist codes with length $n$, rate $R$ and minimum distance $\delta$
that are locally testable with $quasipolylog(n)$
query complexity.

2. Local Correction: For all positive $\epsilon$, for all $\delta < 1/2$ sufficiently
large, and all $R < (1-\epsilon) R_{GV}(\delta)$, there exist codes with length $n$,
rate $R$ and minimum distance $\delta$ that are locally correctable
from $\frac{\delta}{2} - o(1)$ fraction errors with $O(n^{\epsilon})$ query complexity.

Furthermore, these codes have an efficient randomized construction,
and the local testing and local correction algorithms can
be made to run in time polynomial in the query complexity.
Our results on locally correctable codes also immediately give locally decodable codes with the same parameters.

Our local testing result is obtained by combining Thommesen's random concatenation technique
and the best known locally testable codes.
Our local correction result, which is significantly more involved,
also uses random concatenation, along with a number of further ideas:
the Guruswami-Sudan-Indyk list decoding strategy for concatenated codes,
Alon-Edmonds-Luby distance amplification, and the
local list-decodability, local list-recoverability and local testability
of Reed-Muller codes.
Curiously, our final local correction algorithms go via local
list-decoding and local testing algorithms; this seems
to be the first time local testability is used in the
construction of a locally correctable code.