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Robust Locally Testable Codes and Products of Codes [chapter]

Eli Ben-Sasson, Madhu Sudan
2004 Lecture Notes in Computer Science  
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]

Eli Ben-Sasson, Madhu Sudan
2004 arXiv   pre-print
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

Eli Ben-Sasson, Madhu Sudan
2006 Random structures & algorithms (Print)  
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]

Paul Valiant
2005 Lecture Notes in Computer Science  
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]

Lior Eldar
2017 arXiv   pre-print
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

Oded Goldreich, Or Meir
2012 Information Processing Letters  
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]

Michael Viderman
2011 arXiv   pre-print
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

Irit Dinur, Prahladh Harsha, Tali Kaufman, Noga Ron-Zewi, Michael Wagner
2018 Innovations in Theoretical Computer Science  
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]

Irit Dinur, Madhu Sudan, Avi Wigderson
2006 Lecture Notes in Computer Science  
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

Michael Viderman, Michael Wagner
2018 International Workshop on Approximation Algorithms for Combinatorial Optimization  
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]

Don Coppersmith, Atri Rudra
2005 Electronic colloquium on computational complexity  
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]

Gleb Kalachev, Pavel Panteleev
2022 arXiv   pre-print
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

Michael Viderman
2013 2013 IEEE 54th Annual Symposium on Foundations of Computer Science  
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

Or Meir
2012 Information Processing Letters  
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]

Lior Eldar, Aram W. Harrow
2016 arXiv   pre-print
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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