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Exponential lower bound for 2query locally decodable codes via a quantum argument
 Journal of Computer and System Sciences
, 2003
"... Abstract A locally decodable code encodes nbit strings x in mbit codewords C(x) in such a way that one can recover any bit xi from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2 classical queries require exponential length: m = 2 ..."
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Cited by 123 (18 self)
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Abstract A locally decodable code encodes nbit strings x in mbit codewords C(x) in such a way that one can recover any bit xi from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2 classical queries require exponential length: m = 2 \Omega (n). Previously this was known only for linear codes (Goldreich et al. 02). The
Robust PCPs of Proximity, Shorter PCPs and Applications to Coding
 in Proc. 36th ACM Symp. on Theory of Computing
, 2004
"... We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) ..."
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Cited by 80 (25 self)
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We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) Boolean queries.
Locally Testable Codes and PCPs of AlmostLinear Length
, 2002
"... Locally testable codes are errorcorrecting codes that admit very efficient codeword tests. Specifically, using ..."
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Cited by 70 (19 self)
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Locally testable codes are errorcorrecting codes that admit very efficient codeword tests. Specifically, using
Lower Bounds for Linear Locally Decodable Codes and Private Information Retrieval
, 2002
"... We prove that if a linear errorcorrecting code C : f0; 1g is such that a bit of the message can be probabilistically reconstructed by looking at two entries of a corrupted codeword, then 2\Omega\Gamma n) . We also present several extensions of this result. ..."
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Cited by 51 (5 self)
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We prove that if a linear errorcorrecting code C : f0; 1g is such that a bit of the message can be probabilistically reconstructed by looking at two entries of a corrupted codeword, then 2\Omega\Gamma n) . We also present several extensions of this result.
Extractors: Optimal up to Constant Factors
 STOC'03
, 2003
"... This paper provides the first explicit construction of extractors which are simultaneously optimal up to constant factors in both seed length and output length. More precisely, for every n, k, our extractor uses a random seed of length O(log n) to transform any random source on n bits with (min)ent ..."
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Cited by 51 (12 self)
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This paper provides the first explicit construction of extractors which are simultaneously optimal up to constant factors in both seed length and output length. More precisely, for every n, k, our extractor uses a random seed of length O(log n) to transform any random source on n bits with (min)entropy k, into a distribution on (1 − α)k bits that is ɛclose to uniform. Here α and ɛ can be taken to be any positive constants. (In fact, ɛ can be almost polynomially small). Our improvements are obtained via three new techniques, each of which may be of independent interest. The first is a general construction of mergers [22] from locally decodable errorcorrecting codes. The second introduces new condensers that have constant seed length (and retain a constant fraction of the minentropy in the random source). The third is a way to augment the “winwin repeated condensing” paradigm of [17] with error reduction techniques like [15] so that the our constant seedlength condensers can be used without error accumulation.
On WorstCase to AverageCase Reductions for NP Problems
 IN PROCEEDINGS OF THE 44TH IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2003
"... We show that if an NPcomplete problem has a nonadaptive selfcorrector with respect to a samplable distribution then coNP is contained in AM/poly and the polynomial hierarchy collapses to the third level. Feigenbaum and Fortnow show the same conclusion under the stronger assumption that an NPcompl ..."
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Cited by 51 (5 self)
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We show that if an NPcomplete problem has a nonadaptive selfcorrector with respect to a samplable distribution then coNP is contained in AM/poly and the polynomial hierarchy collapses to the third level. Feigenbaum and Fortnow show the same conclusion under the stronger assumption that an NPcomplete problem has a nonadaptive random selfreduction. Our result
Some Applications of Coding Theory in Computational Complexity
, 2004
"... Errorcorrecting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locallytestable and locallydecodable errorcorrecting codes, and their applications to complexity theory ..."
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Cited by 49 (2 self)
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Errorcorrecting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locallytestable and locallydecodable errorcorrecting codes, and their applications to complexity theory and to cryptography.
Towards 3query locally decodable codes of subexponential length
 In Proc. of the 39th ACM Symposium on Theory of Computing (STOC
, 2007
"... A qquery Locally Decodable Code (LDC) encodes an nbit message x as an Nbit codeword C(x), such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. We give new construc ..."
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Cited by 48 (5 self)
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A qquery Locally Decodable Code (LDC) encodes an nbit message x as an Nbit codeword C(x), such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. We give new constructions of three query LDCs of vastly shorter length than that of previous constructions. Specifically, given any Mersenne prime p = 2 t − 1, we design three query LDCs of length N = exp � n 1/t �, for every n. Based on the largest known Mersenne prime, this translates to a length of less than exp n 10−7� compared to exp � n1/2 � in the previous constructions. It has often been conjectured that there are infinitely many Mersenne�primes. Under this conjecture, our constructions yield three query locally decodable codes of length N = exp n O
Robust InformationTheoretic Private Information Retrieval
 Proc. of the 28th International Colloquium on Automata, Languages and Programming, volume 2076 of Lecture Notes in Computer Science
, 2002
"... A Private Information Retrieval (PIR) protocol allows a user to retrieve a data item of its choice from a database, such that the servers storing the database do not gain information on the identity of the item being retrieved. PIR protocols were studied in depth since the subject was introduced in ..."
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Cited by 42 (5 self)
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A Private Information Retrieval (PIR) protocol allows a user to retrieve a data item of its choice from a database, such that the servers storing the database do not gain information on the identity of the item being retrieved. PIR protocols were studied in depth since the subject was introduced in Chor, Goldreich, Kushilevitz, and Sudan 1995. The standard definition of PIR protocols raises a simple question  what happens if some of the servers crash during the operation? How can we devise a protocol which still works in the presence of crashing servers? Current systems do not guarantee availability of servers at all times for many reasons, e.g., crash of server or communication problems. Our purpose is to design robust PIR protocols, i.e., protocols which still work correctly even if only k out of # servers are available during the protocols' operation (the user does not know in advance which servers are available). We present various robust PIR protocols giving different tradeoffs between the different parameters. These protocols are incomparable, i.e., for different values of n and k we will get better results using different protocols. We first present a generic transformation from regular PIR protocols to robust PIR protocols, this transformation is important since any improvement in the communication complexity of regular PIR protocol will immediately implicate improvement in the robust PIR protocol communication. We also present two specific robust PIR protocols. Finally, we present robust PIR protocols which can tolerate Byzantine servers, i.e., robust PIR protocols which still work in the presence of malicious servers or servers with corrupted or obsolete databases. 1
3Query Locally Decodable Codes of Subexponential Length
, 2008
"... Locally Decodable Codes (LDC) allow one to decode any particular symbol of the input message by making a constant number of queries to a codeword, even if a constant fraction of the codeword is damaged. In a recent work [Yek08] Yekhanin constructs a log n log log n 3query LDC with subexponential l ..."
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Cited by 36 (2 self)
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Locally Decodable Codes (LDC) allow one to decode any particular symbol of the input message by making a constant number of queries to a codeword, even if a constant fraction of the codeword is damaged. In a recent work [Yek08] Yekhanin constructs a log n log log n 3query LDC with subexponential length of size exp(exp(O ())). However, this construction requires a conjecture that there are infinitely many Mersenne primes. In this paper we give the first unconditional constant query LDC construction with subexponantial codeword length. In addition our construction reduces codeword length. We give construction of 3query LDC with codeword length exp(exp(O ( √ log n log log n))). Our construction also could be extended to higher number of queries. We give a 2rquery LDC with length of exp(exp(O ( r √ log n(log log n) r−1))). 1