Probabilistic checking of proofs; a new characterization of NP

Arora, Sanjeev; Safra, Shmuel

doi:10.1109/sfcs.1992.267824

Cited by 332 publications

(418 citation statements)

References 20 publications

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“…Gopalan et al [26] showed that tensor products preserve the list-decoding properties, i.e., if C is list-decodable in polynomial time then C m is list-decodable in polynomial time. 4 Our contribution to this question is as follows. We show (Proposition A.2) that if C is decodable from a constant fraction of errors in linear time then C m is decodable from a constant fraction of errors in linear time.…”

Section: Efficient Encoding and Decodingmentioning

confidence: 99%

A combination of testability and decodability by tensor products

Viderman

2013

Random Struct. Alg.

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Ben‐Sasson and Sudan (RSA 2006) showed that taking the repeated tensor product of linear codes with very large distance results in codes that are locally testable. Due to the large distance requirement the associated tensor products could be applied only over sufficiently large fields. Then Meir (SICOMP 2009) used this result to present a combinatorial construction of locally testable codes with largest known rate. As a consequence, this construction was obtained over sufficiently large fields. 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. Consequently, the construction of Meir can be taken over any field, including the binary field. Moreover, a combination of our result with the result of Spielman (IEEE IT, 1996) implies a construction of linear codes (over any field) that combine the following properties: have constant rate and constant relative distance; have blocklength n and are testable with nϵ queries, for any constant ϵ > 0; linear time encodable and linear‐time decodable from a constant fraction of errors. Furthermore, a combination of our result with the result of Guruswami et al. (STOC 2009) implies a similar corollary for list‐decodable codes. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 572–598, 2015

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Section: Efficient Encoding and Decodingmentioning

confidence: 99%

A combination of testability and decodability by tensor products

Viderman

2013

Random Struct. Alg.

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“…2 However, let S be the random variables which Alice and Bob produce in Step 2. It will follow from Corollary 4.3 that…”

Section: For Every Indexmentioning

confidence: 99%

“…Then P U i |W − P U i ≥ ε and the lemma states that Pr[W ] ≤ 2 −kε 2 , which is a version of Chernoff's inequality (note that this implies that Lemma 4.1 is almost tight; see, for example, [22]). 2 The values d i can be chosen independently, but not the values of the x i or the y i . We quickly explain why this is impossible in general.…”

Section: Conditioned Distributionsmentioning

confidence: 99%

“…Therefore, a bit which is contained in both x and y must be considered when generating the values of x i and y i . 2) which is the form we use later. The proof of Lemma 4.1 uses the relative entropy D(P S P T ), which, for two distributions P S and P T over the same set S, is defined as…”

Section: Conditioned Distributionsmentioning

confidence: 99%

“…However, later a counterexample to this conjecture was given [18]. 1 Related work Because of the PCP-Theorem and its application to hardness of approximation [14,2,1], people became interested to know whether the winning probability would at least decrease exponentially in n. Various papers give upper bounds on the winning probability of a game which is repeated n times in parallel [8,13,25,34]; but all of them fall short to give a result of the desired form. Raz then gave a very strong result in [32], showing that the winning probability is at mostv n/log(s) for some constantv < 1 depending only on v, and where s is the total number of answers Alice and Bob may give in the initial game.…”

Section: Introductionmentioning

confidence: 99%

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