Limits on the rate of locally testable affine-invariant codes

A linear code is said to be affine-invariant if the coordinates of the code can be viewed as a vector space and the code is invariant under an affine transformation of the coordinates. A code is said to be locally testable if proximity of a received word to the code can be tested by querying the received word in a few coordinates. Locally testable codes have played a critical role in the construction of probabilistically checkable proofs and most such codes originate from simple affine invariant codes (in particular the Reed-Muller codes). Furthermore it turns out that the local testability of these codes can really be attributed to their affine-invariance. It was hoped that by studying broader classes of affine-invariant codes, one may find nicer, or simpler, locally testable codes, and in particular improve (significantly) on the rate achieved by Reed-Muller codes. In this work we show that low-rate is an inherent limitation of affine-invariant codes. We show that any k-query affine...

• Type: Journal
• Year: 2010
• Where: ECCC
• Authors: Eli Ben-Sasson, Madhu Sudan
• Keywords: Affine-invariant Codes, Code, ECCC 2010, Testable Codes