Locally Testable Codes and Cayley Graphs

TL;DR

This paper characterizes locally testable codes using Cayley graphs over ^h, linking code properties to graph embeddings and eigenvalues, and extends previous results in the field.

Contribution

It provides two new characterizations of -linear locally testable codes via Cayley graphs, connecting code properties to graph embeddings and spectral features.

Findings

Cayley graph generators are larger than h with no short linear dependencies, embedding into  with constant distortion.

Locally testable codes correspond to Cayley graphs with many eigenvalues near 1, explaining all large eigenvalues.

The results extend and converse previous theorems relating codes and Cayley graph properties.

Abstract

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 $\ell_1$ 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 $\ell_1$. \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}