{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9962","name":"On Axis-Parallel Tests for Tensor Product Codes","abstract":"Many low-degree tests examine the input function via its restrictions to random hyperplanes of a certain dimension. Examples include the line-vs-line (Arora, Sudan 2003), plane-vs-plane (Raz, Safra 1997), and cube-vs-cube (Bhangale, Dinur, Livni 2017) tests.\r\n\r\nIn this paper we study tests that only consider restrictions along axis-parallel hyperplanes, which have been studied by Polishchuk and Spielman (1994) and Ben-Sasson and Sudan (2006). While such tests are necessarily \"weaker\", they work for a more general class of codes, namely tensor product codes. Moreover, axis-parallel tests play a key role in constructing LTCs with inverse polylogarithmic rate and short PCPs (Polishchuk, Spielman 1994; Ben-Sasson, Sudan 2008; Meir 2010). We present two results on axis-parallel tests.\r\n\r\n(1) Bivariate low-degree testing with low-agreement. We prove an analogue of the Bivariate Low-Degree Testing Theorem of Polishchuk and Spielman in the low-agreement regime, albeit with much larger field size. Namely, for the 2-wise tensor product of the Reed-Solomon code, we prove that for sufficiently large fields, the 2-query variant of the axis-parallel line test (row-vs-column test) works for arbitrarily small agreement. Prior analyses of axis-parallel tests assumed high agreement, and no results for such tests in the low-agreement regime were known.\r\n\r\nOur proof technique deviates significantly from that of Polishchuk and Spielman, which relies on algebraic methods such as Bezout's Theorem, and instead leverages a fundamental result in extremal graph theory by Kovari, Sos, and Turan. To our knowledge, this is the first time this result is used in the context of low-degree testing.\r\n\r\n\r\n(2) Improved robustness for tensor product codes. Robustness is a strengthening of local testability that underlies many applications. We prove that the axis-parallel hyperplane test for the m-wise tensor product of a linear code with block length n and distance d is Omega(d^m\/n^m)-robust. This improves on a theorem of Viderman (2012) by a factor of 1\/poly(m). While the improvement is not large, we believe that our proof is a notable simplification compared to prior work.","keywords":["tensor product codes","locally testable codes","low-degree testing","extremal graph theory"],"author":[{"@type":"Person","name":"Chiesa, Alessandro","givenName":"Alessandro","familyName":"Chiesa"},{"@type":"Person","name":"Manohar, Peter","givenName":"Peter","familyName":"Manohar"},{"@type":"Person","name":"Shinkar, Igor","givenName":"Igor","familyName":"Shinkar"}],"position":39,"pageStart":"39:1","pageEnd":"39:22","dateCreated":"2017-08-11","datePublished":"2017-08-11","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Chiesa, Alessandro","givenName":"Alessandro","familyName":"Chiesa"},{"@type":"Person","name":"Manohar, Peter","givenName":"Peter","familyName":"Manohar"},{"@type":"Person","name":"Shinkar, Igor","givenName":"Igor","familyName":"Shinkar"}],"copyrightYear":"2017","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.APPROX-RANDOM.2017.39","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6284","volumeNumber":81,"name":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX\/RANDOM 2017)","dateCreated":"2017-08-11","datePublished":"2017-08-11","editor":[{"@type":"Person","name":"Jansen, Klaus","givenName":"Klaus","familyName":"Jansen"},{"@type":"Person","name":"Rolim, Jos\u00e9 D. P.","givenName":"Jos\u00e9 D. P.","familyName":"Rolim"},{"@type":"Person","name":"Williamson, David P.","givenName":"David P.","familyName":"Williamson"},{"@type":"Person","name":"Vempala, Santosh S.","givenName":"Santosh S.","familyName":"Vempala"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9962","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6284"}}}