Locally testable codes and cayley graphs

  • Authors:

  • Parikshit Gopalan;Salil Vadhan;Yuan Zhou

  • Affiliations:

  • Microsoft Research Silicon Valley, Mountain View, CA, USA;Harvard University, Cambridge, MA, USA;Carnegie Mellon University, Pittsburgh, PA, USA

  • Venue:
  • Proceedings of the 5th conference on Innovations in theoretical computer science
  • Year:
  • 2014

Quantified Score

Hi-index 0.00

Visualization

Abstract

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.