Tolerant Linearity Testing and Locally Testable Codes

Abstract. 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) = 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 Zn 2 are locally testable.

• Type: Conference
• Year: 2009
• Where: APPROX
• Authors: Swastik Kopparty, Shubhangi Saraf
• Keywords: Algorithms, APPROX 2009, Linear Codes, Linearity, Tolerant Linearity Testing