We give two new characterizations of (\F_2-linear) locally testable
error-correcting codes in terms of Cayley graphs over \F_2^h:
\begin{enumerate} \item A locally testable code is equivalent to a Cayley
graph over \F_2^h whose set of generators is significantly larger than h
and has no short linear dependencies, but yields a shortest-path metric that
embeds into โ1โ with constant distortion. This extends and gives a
converse to a result of Khot and Naor (2006), which showed that codes with
large dual distance imply Cayley graphs that have no low-distortion embeddings
into โ1โ.
\item A locally testable code is equivalent to a Cayley graph over \F_2^h
that has significantly more than h eigenvalues near 1, which have no short
linear dependencies among them and which "explain" all of the large
eigenvalues. This extends and gives a converse to a recent construction of
Barak et al. (2012), which showed that locally testable codes imply Cayley
graphs that are small-set expanders but have many large eigenvalues.
\end{enumerate}Comment: 22 page