Absolutely Sound Testing of Lifted Codes

In this work we present a strong analysis of the testability of a broad, and to date the most interesting known, class of “affine-invariant” codes. Affine-invariant codes are codes whose coordinates are associated with a vector space and are invariant under affine transformations of the coordinate space. Affine-invariant linear codes form a natural abstraction of algebraic properties such as linearity and low-degree, which have been of significant interest in theoretical computer science in the past. The study of affine-invariance is motivated in part by its relationship to property testing: Affine-invariant linear codes tend to be locally testable under fairly minimal and almost necessary conditions. Recent works by Ben-Sasson et al. (CCC 2011) and Guo et al. (ITCS 2013) have introduced a new class of affine-invariant linear codes based on an operation called “lifting”. Given a base code over a t-dimensional space, its m-dimensional lift consists of all words whose restriction to every t-dimensional affine subspace is a codeword of the base code. Lifting not only captures the most familiar codes, which can be expressed as lifts of low-degree polynomials, it also yields new codes when lifting “medium-degree” polynomials whose rate is better than that of corresponding polynomial codes, and all other combinatorial qualities are no worse. In this work we show that codes derived from lifting are also testable in an “absolutely sound” way. Specifically, we consider the natural test: Pick a random affine subspace of base dimension and verify that a given word is a codeword of the base code when restricted to the chosen subspace. We show that this test accepts codewords with probability one, while rejecting words at constant distance from the code with constant probability (depending only on the alphabet size). This work thus extends the results of Bhattacharyya et al. (FOCS 2010) and Haramaty et al. (FOCS 2011), while giving concrete new codes of higher rate that have absolutely sound testers.