License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.39
URN: urn:nbn:de:0030-drops-75882
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7588/
Chiesa, Alessandro ;
Manohar, Peter ;
Shinkar, Igor
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.
In 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.
(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.
Our 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.
(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.
BibTeX - Entry
@InProceedings{chiesa_et_al:LIPIcs:2017:7588,
author = {Alessandro Chiesa and Peter Manohar and Igor Shinkar},
title = {{On Axis-Parallel Tests for Tensor Product Codes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {39:1--39:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Klaus Jansen and Jos{\'e} D. P. Rolim and David Williamson and Santosh S. Vempala},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7588},
URN = {urn:nbn:de:0030-drops-75882},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.39},
annote = {Keywords: tensor product codes, locally testable codes, low-degree testing, extremal graph theory}
}
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Keywords: |
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tensor product codes, locally testable codes, low-degree testing, extremal graph theory |
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Collection: |
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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017) |
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Issue Date: |
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2017 |
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Date of publication: |
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11.08.2017 |
