Robust Locally Testable Codes and Products of Codes

TL;DR

This paper introduces the concept of robust local testability for codes, relates it to code products, and demonstrates how tensor product codes can be tested efficiently with few queries, advancing the theory of error-correcting codes.

Contribution

It defines robust local testability, connects it to code products, and provides a generic construction of efficiently testable codes from any linear code with good distance.

Findings

Robust local testability relates to code products and can be composed simply.

Tensor product codes can be tested robustly and locally using adapted techniques.

Constructs codes with inverse polynomial rate that are testable with poly-logarithmic queries.

Abstract

We continue the investigation of locally testable codes, i.e., error-correcting codes for whom membership of a given word in the code can be tested probabilistically by examining it in very few locations. We give two general results on local testability: First, motivated by the recently proposed notion of {\em robust} probabilistically checkable proofs, we introduce the notion of {\em robust} local testability of codes. We relate this notion to a product of codes introduced by Tanner, and show a very simple composition lemma for this notion. Next, we show that codes built by tensor products can be tested robustly and somewhat locally, by applying a variant of a test and proof technique introduced by Raz and Safra in the context of testing low-degree multivariate polynomials (which are a special case of tensor codes). Combining these two results gives us a generic construction of codes of inverse polynomial rate, that are testable with poly-logarithmically many queries. We note these locally testable tensor codes can be obtained from {\em any} linear error correcting code with good distance. Previous results on local testability, albeit much stronger quantitatively, rely heavily on algebraic properties of the underlying codes.