  In this work, we present the first local-decoding algorithm for expander
codes. This yields a new family of constant-rate codes that can recover from a
constant fraction of errors in the codeword symbols, and where any symbol of
the codeword can be recovered with high probability by reading $N^\epsilon$
symbols from the corrupted codeword, where $N$ is the block-length of the code.
  Expander codes, introduced by Sipser and Spielman, are formed from an
expander graph $G = (V,E)$ of degree $d$, and an inner code of block-length $d$
over an alphabet $\Sigma$. Each edge of the expander graph is associated with a
symbol in $\Sigma$. A string in $\Sigma^{E}$ will be a codeword if for each
vertex in $V$, the symbols on the adjacent edges form a codeword in the inner
code.
  We show that if the inner code has a smooth reconstruction algorithm in the
noiseless setting, then the corresponding expander code has an efficient
local-correction algorithm in the noisy setting. Instantiating our construction
with inner codes based on finite geometries, we obtain novel locally decodable
codes with rate approaching one. This provides an alternative to the
multiplicity codes of Kopparty, Saraf and Yekhanin (STOC '11) and the lifted
codes of Guo, Kopparty and Sudan (ITCS '13).
