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URN: urn:nbn:de:0030-drops-68524
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Robust Multiplication-Based Tests for Reed-Muller Codes

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Abstract

We consider the following multiplication-based tests to check if a given function f: F^n_q -> F_q is the evaluation of a degree-d polynomial over F_q for q prime. Test_{e,k}: Pick P_1,...,P_k independent random degree-e polynomials and accept iff the function f P_1 ... P_k is the evaluation of a degree-(d + ek) polynomial. We prove the robust soundness of the above tests for large values of e, answering a question of Dinur and Guruswami (FOCS 2013). Previous soundness analyses of these tests were known only for the case when either e = 1 or k = 1. Even for the case k = 1 and e > 1, earlier soundness analyses were not robust. We also analyze a derandomized version of this test, where (for example) the polynomials P_1 ,... , P_k can be the same random polynomial P. This generalizes a result of Guruswami et al. (STOC 2014). One of the key ingredients that go into the proof of this robust soundness is an extension of the standard Schwartz-Zippel lemma over general finite fields F_q, which may be of independent interest.

BibTeX - Entry

@InProceedings{harsha_et_al:LIPIcs:2016:6852,
  author =	{Prahladh Harsha and Srikanth Srinivasan},
  title =	{{Robust Multiplication-Based Tests for Reed-Muller Codes}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{17:1--17:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Akash Lal and S. Akshay and Saket Saurabh and Sandeep Sen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6852},
  URN =		{urn:nbn:de:0030-drops-68524},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.17},
  annote =	{Keywords: Polynomials over finite fields, Schwartz-Zippel lemma, Low degree testing, Low degree long code}
}

Keywords: Polynomials over finite fields, Schwartz-Zippel lemma, Low degree testing, Low degree long code
Seminar: 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)
Issue date: 2016
Date of publication: 2016


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