Locally testable codes and cayley graphs

We give two new characterizations of ( 2-linear, smooth) locally testable error-correcting codes in terms of Cayley graphs over Fh2: A locally testable code is equivalent to a Cayley graph over h2 whose set of generators is significantly larger than h and has no short linear dependencies, bbut yields a shortest-path metric that embeds into l 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 l . A locally testable code is equivalent to a Cayley graph over Fh2 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.