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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 56 (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
Two Query PCP with SubConstant Error
, 2008
"... We show that the N PComplete language 3SAT has a PCP verifier that makes two queries to a proof of almostlinear size and achieves subconstant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer ..."
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Cited by 51 (5 self)
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We show that the N PComplete language 3SAT has a PCP verifier that makes two queries to a proof of almostlinear size and achieves subconstant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer to the second query. Previously, by the parallel repetition theorem, there were PCP Theorems with twoquery projection tests, but only (arbitrarily small) constant error and polynomial size [29]. There were also PCP Theorems with subconstant error and almostlinear size, but a constant number of queries that is larger than 2 [26]. As a corollary, we obtain a host of new results. In particular, our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem. A partial list includes the following: 1. 3SAT cannot be efficiently approximated to within a factor of 7 8 + o(1), unless P = N P. This holds even under almostlinear reductions. Previously, the best known N Phardness
On the locality of codeword symbols
 IEEE Trans. Inform. Theory
, 2012
"... Consider a linear [n, k, d]q code C. We say that that ith coordinate of C has locality r, if the value at this coordinate can be recovered from accessing some other r coordinates of C. Data storage applications require codes with small redundancy, low locality for information coordinates, large dis ..."
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Cited by 49 (2 self)
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Consider a linear [n, k, d]q code C. We say that that ith coordinate of C has locality r, if the value at this coordinate can be recovered from accessing some other r coordinates of C. Data storage applications require codes with small redundancy, low locality for information coordinates, large distance, and low locality for parity coordinates. In this paper we carry out an indepth study of the relations between these parameters. We establish a tight bound for the redundancy n−k in terms of the message length, the distance, and the locality of information coordinates. We refer to codes attaining the bound as optimal. We prove some structure theorems about optimal codes, which are particularly strong for small distances. This gives a fairly complete picture of the tradeoffs between codewords length, worstcase distance and locality of information symbols. We then consider the locality of parity check symbols and erasure correction beyond worst case distance for optimal codes. Using our structure theorem, we obtain a tight bound for the locality of parity symbols possible in such codes for a broad class of parameter settings. We prove that there is a tradeoff between having good locality for parity checks and the ability to correct erasures beyond the minimum distance. 1
Improving the Robustness of Private Information Retrieval
 In Proceedings of IEEE Security and Privacy Symposium
, 2007
"... Since 1995, much work has been done creating protocols for private information retrieval (PIR). Many variants of the basic PIR model have been proposed, including such modifications as computational vs. informationtheoretic privacy protection, correctness in the face of servers that fail to respond ..."
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Cited by 46 (16 self)
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Since 1995, much work has been done creating protocols for private information retrieval (PIR). Many variants of the basic PIR model have been proposed, including such modifications as computational vs. informationtheoretic privacy protection, correctness in the face of servers that fail to respond or that respond incorrectly, and protection of sensitive data against the database servers themselves. In this paper, we improve on the robustness of PIR in a number of ways. First, we present a Byzantinerobust PIR protocol which provides informationtheoretic privacy protection against coalitions of up to all but one of the responding servers, improving the previous result by a factor of 3. In addition, our protocol allows for more of the responding servers to return incorrect information while still enabling the user to compute the correct result. We then extend our protocol so that queries have informationtheoretic protection if a limited number of servers collude, as before, but still retain computational protection if they all collude. We also extend the protocol to provide informationtheoretic protection to the contents of the database against collusions of limited numbers of the database servers, at no additional communication cost or increase in the number of servers. All of our protocols retrieve a block of data with communication cost only O(ℓ) times the size of the block, where ℓ is the number of servers. Finally, we discuss our implementation of these protocols, and measure their performance in order to determine their practicality. 1
A hypercontractive inequality for matrixvalued functions with applications to quantum computing and LDCs
"... The BonamiBeckner hypercontractive inequality is a powerful tool in Fourier analysis of realvalued functions on the Boolean cube. In this paper we present a version of this inequality for matrixvalued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and ..."
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Cited by 39 (3 self)
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The BonamiBeckner hypercontractive inequality is a powerful tool in Fourier analysis of realvalued functions on the Boolean cube. In this paper we present a version of this inequality for matrixvalued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode n classical bits into m qubits, in such a way that each set of k bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if m<0.7n, then the success probability is exponentially small in k. This result may be viewed as a direct product version of Nayak’s quantum random access code bound. It in turn implies strong direct product theorems for the oneway quantum communication complexity of Disjointness and other problems. Second, we prove that errorcorrecting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first “nonquantum” proof of a result originally derived by Kerenidis and de Wolf using quantum information theory.
A Note on Yekhanin’s Locally Decodable Codes
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 16 (2007)
, 2007
"... Locally Decodable codes(LDC) support decoding of any particular symbol of the input message by reading constant number of symbols of the codeword, even in presence of constant fraction of errors. In a recent breakthrough [9], Yekhanin constructedquery LDCs that hugely improve over earlier construct ..."
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Cited by 19 (0 self)
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Locally Decodable codes(LDC) support decoding of any particular symbol of the input message by reading constant number of symbols of the codeword, even in presence of constant fraction of errors. In a recent breakthrough [9], Yekhanin constructedquery LDCs that hugely improve over earlier constructions. Specifically, for a Mersenne prime, binary LDCs of length for infinitely many were obtained. Using the largest known Mersenne prime, this implies LDCs of length less than. Assuming infinitude of Mersenne primes, the construction yields LDCs of length for infinitely many. Inspired by [9], we constructquery binary LDCs with same parameters from Mersenne primes. While all the main technical tools are borrowed from [9], we give a selfcontained simple construction of LDCs. Our bounds do not improve over [9], and have worse soundness of the decoder. However the LDCs are simpler and generalize naturally to prime fields other than. The LDCs presented also translate directly in to three server Private Information Retrieval(PIR) protocols with communication! complexities for a database of size, starting with a Mersenne prime.
Sparse random linear codes are locally decodable and testable
 in Proc. 40th STOC
, 2007
"... We show that random sparse binary linear codes are locally testable and locally decodable (under any linear encoding) with constant queries (with probability tending to one). By sparse, we mean that the code should have only polynomially many codewords. Our results are the first to show that local d ..."
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Cited by 17 (6 self)
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We show that random sparse binary linear codes are locally testable and locally decodable (under any linear encoding) with constant queries (with probability tending to one). By sparse, we mean that the code should have only polynomially many codewords. Our results are the first to show that local decodability and testability can be found in random, unstructured, codes. Previously known locally decodable or testable codes were either classical algebraic codes, or new ones constructed very carefully. We obtain our results by extending the techniques of Kaufman and Litsyn [11] who used the MacWilliams Identities to show that “almostorthogonal ” binary codes are locally testable. Their definition of almost orthogonality expected codewords to disagree in n 2 ± O( √ n) coordinates in codes of block length n. The only families of codes known to have this property were the dualBCH codes. We extend their techniques, and simplify them in the process, to include codes of distance at least n 2 −O(n1−γ) for any γ> 0, provided the number of codewords is O(n t) for some constant t. Thus our results derive the local testability of linear codes from the classical coding theory parameters, namely the rate and the distance of the codes. More significantly, we show that this technique can also be used to prove the “selfcorrectability ” of sparse codes of sufficiently large distance. This allows us to show that random linear codes under linear encoding functions are locally decodable. This ought to be surprising in that the definition of a code doesn’t specify the encoding function used! Our results effectively say that any linear function of the bits of the codeword can be locally decoded in this case.
New constructions for queryefficient locally decodable codes of subexponential length
 IEICE Transactions on Information and Systems
"... is an errorcorrecting code that encodes each message ⃗x = (x1, x2,...,xn) ∈ Fn q to a codeword C(⃗x) ∈ FNq and has the following property: For any ⃗y ∈ FN q such that d(⃗y, C(⃗x)) ≤ δN and each 1 ≤ i ≤ n, the symbol xi of ⃗x can be recovered with probability at least 1−ε by a randomized decoding ..."
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Cited by 16 (0 self)
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is an errorcorrecting code that encodes each message ⃗x = (x1, x2,...,xn) ∈ Fn q to a codeword C(⃗x) ∈ FNq and has the following property: For any ⃗y ∈ FN q such that d(⃗y, C(⃗x)) ≤ δN and each 1 ≤ i ≤ n, the symbol xi of ⃗x can be recovered with probability at least 1−ε by a randomized decoding algorithm looking only at k coordinates of ⃗y. The efficiency of a (k, δ, ε)locally decodable code C: Fn q → FN q is measured by the code length N and the Abstract: A (k, δ, ε)locally decodable code C: F n q → FN q number k of queries. For any kquery locally decodable code C: Fn q → FNq, the code length N is conjectured to be exponential of n, i.e., N = exp(nΩ(1)), however, this was disproved. Yekhanin [In Proc. such that of STOC, 2007] showed that there exists a 3query locally decodable code C: Fn 2 → FN2 N = exp(n (1 / log log n) ) assuming that the number of Mersenne primes is infinite. For a 3query locally decodable code C: Fn q → FN q, Efremenko [ECCC Report No.69, 2008] reduced the code length further to N = exp(nO((log log n/log n)1/2)), and also showed that for any integer r> 1, there exists a kquery locally decodable code C: Fn q → FN q such that k ≤ 2r and N = exp(nO((log log n/log n)1−1/r)). In this paper, we present a queryefficient locally decodable code by introducing a technique of “composition of locally decodable codes, ” and show that for any integer r> 1, there exists a kquery locally decodable code C: Fn q → FNq such that k ≤ 3 · 2r−2 and N = exp(nO((log log n/log n)1−1/r)). Keywords: Locally Decodable Codes, SMatching Vectors, SDecoding Polynomials, Composition of Locally Decodable Codes, Perfectly Smooth Decoders, Private Information Retrieval.
Quantum Proofs for Classical Theorems
, 2009
"... Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in classical (nonquantum) areas. In this paper we survey these results and the quantum toolbox they use. ..."
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Cited by 15 (4 self)
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Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in classical (nonquantum) areas. In this paper we survey these results and the quantum toolbox they use.
QUERYEFFICIENT LOCALLY DECODABLE CODES OF SUBEXPONENTIAL LENGTH
, 2013
"... A kquery locally decodable code (LDC) C: Σn → ΓN encodes each message x into a codeword C(x) such that each symbol of x can be probabilistically recovered by querying only k coordinates of C(x), even after a constant fraction of the coordinates has been corrupted. Yekhanin (in J ACM 55:1–16, 2008 ..."
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Cited by 15 (2 self)
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A kquery locally decodable code (LDC) C: Σn → ΓN encodes each message x into a codeword C(x) such that each symbol of x can be probabilistically recovered by querying only k coordinates of C(x), even after a constant fraction of the coordinates has been corrupted. Yekhanin (in J ACM 55:1–16, 2008) constructed a 3query LDC of subexponential length, N = exp(exp(O(logn / log logn))), under the assumption that there are infinitely many Mersenne primes. Efremenko (in Proceedings of the 41st annual ACM symposium on theory of computing, ACM, New York, 2009) constructed a 3query LDC of length N2 = exp(exp(O( log n log logn))) with no assumption, and a 2rquery LDC of length Nr = exp(exp(O ( r logn(log logn)r−1))), for every integer r ≥ 2. Itoh and Suzuki (in IEICE Trans Inform Syst E93D 2:263– 270, 2010) gave a composition method in Efremenko’s framework and