Tolerant Linearity Testing and Locally Testable Codes

10 years 8 months ago
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.
Swastik Kopparty, Shubhangi Saraf
Added 25 May 2010
Updated 25 May 2010
Type Conference
Year 2009
Authors Swastik Kopparty, Shubhangi Saraf
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