Tolerant Linearity Testing and Locally Testable Codes

We study tolerant linearity testing under general distributions. Given groups G and H, a distribution μ on G, and oracle access to a function f : G→ H, we consider the task of approximating the smallest μ-distance of f to a homomorphism h : G→ H, where the μ-distance between f and h is the probability that f(x) 6= h(x) when x is drawn according to the distribution μ. This question is intimately connected to local testability of linear codes. In this work, we give a general sufficient condition on the distribution μ for linearity to be tolerantly testable with a constant number of queries. Using this condition we show that linearity is tolerantly testable for several natural classes of distributions including low bias, symmetric and product distributions. This gives a new and simple proof of a result of Kaufman and Sudan which shows that sparse, unbiased linear codes over Z2 are locally testable.