Some Recent Results on Local Testing of Sparse Linear Codes

We study the local testability of linear codes. We first reformulate this question in the language of tolerant linearity testing under a non-uniform distribution. We then study the question of linearity testing under non-uniform distributions directly, and give a sufficient criterion for linearity to be tolerantly testable under a given distribution. We show that several natural classes of distributions satisfy this criterion (such as product distributions and low Fourier-bias distributions), thus showing that linearity is tolerantly testable under these distributions. This in turn implies that the corresponding codes are locally testable. For the case of random sparse linear codes, we show the testability and decodability of such codes in the presence of very high noise rates. More precisely, we show that any linear code in Fn 2 which is:

• Type: Journal
• Year: 2010
• Where: PROPERTYTESTING
• Authors: Swastik Kopparty, Shubhangi Saraf
• Keywords: Linear Codes, Linearity, Non-uniform Distributions, PROPERTYTESTING 2010, Theoretical Computer Science