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 $f(x) \ne h(x)$ 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 are locally testable.