A combination of testability and decodability by tensor products1. Issue 3 (12th April 2013)

• Record Type: Journal Article
• Authors: Viderman, Michael
• Is Part Of: Random structures & algorithms. Volume 46:Issue 3(2015)
• Page Start: 572
• Page End: 598
• Publication Date: 2013-04-12
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20498
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Ben‐Sasson and Sudan (RSA 2006) showed that taking the repeated tensor product of linear codes with very large distance results in codes that are locally testable. Due to the large distance requirement the associated tensor products could be applied only over sufficiently large fields. Then Meir (SICOMP 2009) used this result to present a combinatorial construction of locally testable codes with largest known rate. As a consequence, this construction was obtained over sufficiently large fields.</p> <p>In this paper we improve the result of Ben‐Sasson and Sudan and show that for <italic>any</italic> linear codes the associated tensor products are locally testable. Consequently, the construction of Meir can be taken over any field, including the binary field.</p> <p>Moreover, a combination of our result with the result of Spielman (IEEE IT, 1996) implies a construction of linear codes (over any field) that combine the following properties: <list id="rsa20498-list-0001" list-type="bullet"><list-item id="rsa20498-li-0001"><p>have constant rate and constant relative distance;</p></list-item><list-item id="rsa20498-li-0002"><p>have blocklength <italic>n</italic> and are testable with <italic>n</italic><sup><italic>ϵ</italic></sup> queries, for any constant <italic>ϵ</italic> > 0;</p></list-item><list-item id="rsa20498-li-0003"><p>linear time encodable and linear‐time decodable from a constant fraction of errors.</p></list-item></list></p> <p>Furthermore, a combination of our result with the result of Guruswami et al. (STOC 2009) implies a similar corollary for list‐decodable codes. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 572–598, 2015</p> </abstract>