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Robust Locally Testable Codes and Products of Codes
[chapter]
2004
Lecture Notes in Computer Science
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We give two general results on local testability: First, motivated by the recently proposed notion of robust probabilistically checkable proofs, we introduce the notion of robust local testability of codes ...
Next, we show that codes built by tensor products can be tested robustly and somewhat locally, by applying a variant of a test and proof technique introduced by Raz and Safra in the context of testing ...
Acknowledgments We wish to thank Irit Dinur, Oded Goldreich and Prahladh Harsha for valuable discussions. ...
doi:10.1007/978-3-540-27821-4_26
fatcat:2co2b3ihtnhyhhv62hy2ib3azu
Robust Locally Testable Codes and Products of Codes
[article]
2004
arXiv
pre-print
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We give two general results on local testability: First, motivated by the recently proposed notion of robust probabilistically checkable proofs, we introduce the notion of robust local testability of codes ...
Next, we show that codes built by tensor products can be tested robustly and somewhat locally, by applying a variant of a test and proof technique introduced by Raz and Safra in the context of testing ...
Acknowledgments We wish to thank Irit Dinur, Oded Goldreich and Prahladh Harsha for valuable discussions. ...
arXiv:cs/0408066v1
fatcat:blk2ttc37za55ouu3htuigix6i
Robust locally testable codes and products of codes
2006
Random structures & algorithms (Print)
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We give two general results on local testability: First, motivated by the recently proposed notion of robust probabilistically checkable proofs, we introduce the notion of robust local testability of codes ...
Next, we show that codes built by tensor products can be tested robustly and somewhat locally, by applying a variant of a test and proof technique introduced by Raz and Safra in the context of testing ...
Acknowledgments We wish to thank Irit Dinur, Oded Goldreich and Prahladh Harsha for valuable discussions. ...
doi:10.1002/rsa.20120
fatcat:5sw3qjcfwjgp7mdzvvao56doi4
The Tensor Product of Two Codes Is Not Necessarily Robustly Testable
[chapter]
2005
Lecture Notes in Computer Science
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Ben-Sasson and Sudan show that the repeated tensor product of codes leads to a general class of locally testable codes. ...
One question that is not settled by their work is the local testability of a code generated by a single application of the tensor product. ...
Local testability-and in fact robustness-of the product tester for the bivariate polynomial codes is well-studied in the literature on PCPs. ...
doi:10.1007/11538462_40
fatcat:o5sar2s6nre4xfrjugjma2yfxa
Local Hamiltonians with Approximation-Robust Entanglement
[article]
2017
arXiv
pre-print
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2015 pre-print arXiv:1503.02269v3 fatcat:trw7sgt6gfc6nnfwulmjvjpqu4Other Versions
In this study, we indicate that such robustness may be possible after all: We construct an infinite family of O(1)-local Hamiltonians, corresponding to check terms of a quantum error-correcting code with ...
NLTS), and even find such robust forms of entanglement that are useful for some computation. ...
importance of the Tillich-Zémor construction. ...
arXiv:1503.02269v4
fatcat:v5y3bnqcvjainclhvgskoxl3jm
The tensor product of two good codes is not necessarily robustly testable
2012
Information Processing Letters
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Given two codes R and C, their tensor product R ⊗ C consists of all matrices whose rows are codewords of R and whose columns are codewords of C. ...
The product R ⊗ C is said to be robust if for every matrix M that is far from R ⊗ C it holds that the rows and columns of M are far on average from R and C respectively. ...
It is not hard to see that if both R and C are locally testable, and R ⊗ C is robust, then R ⊗ C is locally testable. ...
doi:10.1016/j.ipl.2012.01.007
fatcat:lvqahlqv7jaidnq7kadldlxeti
A Combination of Testability and Decodability by Tensor Products
[article]
2011
arXiv
pre-print
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Ben-Sasson and Sudan (RSA 2006) showed that repeated tensor products of linear codes with a very large distance are locally testable. ...
In this paper we improve the result of Ben-Sasson and Sudan and show that for any linear codes the associated tensor products are locally testable. ...
Acknowledgements We thank Eli Ben-Sasson and Or Meir for helpful discussions. The author thanks also Eli Ben-Sasson for many invaluable discussions regarding the "robustness" concept. ...
arXiv:1105.5806v1
fatcat:hbklsosbmnfsrf5vte2pefik5y
From Local to Robust Testing via Agreement Testing
2018
Innovations in Theoretical Computer Science
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Robust testing is an important component in constructions of locally testable codes and probabilistically checkable proofs as it allows for composition of local tests. ...
Thus, it immediately applies to local testers of lifted codes that query random affine subspaces in F m q , and moreover seems amenable to extension to other families of locally testable codes with expanding ...
We start by describing the notions of local testing, robust testing, and lifted codes.
Local testing and robust testing A code is a subset C ⊆ Σ n . ...
doi:10.4230/lipics.itcs.2019.29
dblp:conf/innovations/DinurHKR19
fatcat:cyk6fyvvhncv7mwluhx6655yca
Robust Local Testability of Tensor Products of LDPC Codes
[chapter]
2006
Lecture Notes in Computer Science
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We show that this test is robust, provided one of the codes is what we refer to as smooth. We show that expander codes and locally-testable codes are smooth. This complements recent examples of P. ...
Valiant [13] and Coppersmith and Rudra [9] of codes whose tensor product is not robustly testable. ...
Robust Locally Testable Codes Locally testable codes, as described in the introduction, are codes for which one can test whether a given word x is in the code by reading only few (randomly chosen) symbols ...
doi:10.1007/11830924_29
fatcat:tdxyigv4srdjdbrfiyi5uheq4e
Explicit Strong LTCs with Inverse Poly-Log Rate and Constant Soundness
2018
International Workshop on Approximation Algorithms for Combinatorial Optimization
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An error-correcting code C ⊆ F n is called (q, )-strong locally testable code (LTC) if there exists a tester that makes at most q queries to the input word. ...
In this work we show that codes presented in the works of Dinur (J.ACM 2007) and Ben-Sasson and Sudan (SICOMP 2005) provide the explicit construction of strong LTCs with the above range of parameters. ...
In general, codes composed as 2-wise tensor products might be not testable [17, 40, 25], however, there are still ways to use such kind of code products to obtain locally testable codes, e.g., [20, 12 ...
doi:10.4230/lipics.approx-random.2018.58
dblp:conf/approx/Viderman18
fatcat:gm7usfjvdbhpvnobhnr77wovge
On the Robust Testability of Product of Codes
[article]
2005
Electronic colloquium on computational complexity
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Ben-Sasson and Sudan in [4] asked if the following test is robust for the tensor product of a code with another code-pick a row (or column) at random and check if the received word restricted to the picked ...
Valiant showed that for general linear codes, the test is not robust [12] . However the question remained open for the tensor product of a code with itself. We resolve this question in the negative. ...
C 1 and C 2 are themselves tensor product of a code [4] . ...
dblp:journals/eccc/ECCC-TR05-104
fatcat:6kzoztbu5ffztnimxbjghthp7a
Two-sided Robustly Testable Codes
[article]
2022
arXiv
pre-print
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We show that this property is related with the robust testability and the agreement testability of the products of codes. ...
This implies that one can obtain pairs of linear codes such that their product and the product of their dual codes are simultaneously robustly testable. ...
Acknowledgment This work was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant 075-15-2020-801). ...
arXiv:2206.09973v1
fatcat:yqb5bwr56fgz5oxf36twmz3cv4
Strong LTCs with Inverse Poly-Log Rate and Constant Soundness
2013
2013 IEEE 54th Annual Symposium on Foundations of Computer Science
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An error-correcting code C ⊆ F n is called (q, )-strong locally testable code (LTC) if there exists a tester that makes at most q queries to the input word. ...
In this work we show that codes presented in the works of Dinur (J.ACM 2007) and Ben-Sasson and Sudan (SICOMP 2005) provide the explicit construction of strong LTCs with the above range of parameters. ...
Acknowledgements The author thanks Or Meir for raising the suggestion to obtain an explicit construction of strong LTCs by applying the arguments of [45, 44] to the codes of [11] . ...
doi:10.1109/focs.2013.43
dblp:conf/focs/Viderman13
fatcat:xs6wjnu77fecphsyofdxsrcxfe
On the rectangle method in proofs of robustness of tensor products
2012
Information Processing Letters
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So far, a few important families of tensor products were shown to be robust, and a counterexample of a product that is not robust was also given. ...
However, a precise characterization of codes whose tensor product is robust is yet unknown. ...
The author also wishes to thank Changyuan Yu for pointing out an error in a previous version of this paper. ...
doi:10.1016/j.ipl.2011.11.007
fatcat:vjpz4r2khreh7fbal4ldfziw4q
Local Hamiltonians Whose Ground States are Hard to Approximate
[article]
2016
arXiv
pre-print
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Our Hamiltonian is based on applying the hypergraph product by Tillich and Zemor to a classical locally testable code. ...
LDPC codes, and the NLTS conjecture due to Freedman and Hastings positing the existence of local Hamiltonians in which any low-energy state is highly-entangled. ...
comments and Alex Lubotzky for pointing out the fundamental importance of the Tillich-Zémor construction. ...
arXiv:1510.02082v3
fatcat:4la4js5ctzf5hokemvt6fsft3a
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