Theory of Computing ------------------- Title : Absolutely Sound Testing of Lifted Codes Authors : Elad Haramaty, Noga Ron-Zewi, and Madhu Sudan Volume : 11 Number : 12 Pages : 299-338 URL : https://theoryofcomputing.org/articles/v011a012 Abstract -------- In this paper 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 in addition these codes 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 work by Ben-Sasson et al. (CCC 2011) and Guo et al. (ITCS 2013) 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 applied to "medium-degree" polynomials whose rate is better than that of the corresponding polynomial codes, and all other combinatorial qualities are no worse. In this paper we show that codes obtained by 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. While this test accepts codewords with probability one, we show that it rejects 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. In particular, we show that there exist codes satisfying the requirements of Barak et al. (FOCS 2012) for constructing small-set expanders with a large number of eigenvalues close to the maximal one, with rate slightly higher than that of the codes used in their work. An extended abstract of this paper appeared in the "Proceedings of the 17th International Workshop on Randomization and Computation (RANDOM'13)," pages 671-682, 2013, http://link.springer.com/chapter/10.1007/978-3-642-40328-6_46