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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.
Random Cayley Graphs and Expanders
 Random Structures Algorithms
, 1997
"... For every 1 ? ffi ? 0 there exists a c = c(ffi) ? 0 such that for every group G of order n, and for a set S of c(ffi) log n random elements in the group, the expected value of the second largest eigenvalue of the normalized adjacency matrix of the Cayley graph X(G;S) is at most (1\Gammaffi). Thi ..."
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Cited by 34 (0 self)
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For every 1 ? ffi ? 0 there exists a c = c(ffi) ? 0 such that for every group G of order n, and for a set S of c(ffi) log n random elements in the group, the expected value of the second largest eigenvalue of the normalized adjacency matrix of the Cayley graph X(G;S) is at most (1\Gammaffi). This implies that almost every such a graph is an "(ffi)expander. For Abelian groups this is essentially tight, and explicit constructions can be given in some cases. Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel. Research supported in part by a U.S.A.Israeli BSF grant. y Department of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel 0 1.
Constructing Small Sets That Are Uniform in Arithmetic Progressions
, 2002
"... this paper also satisfy (A ;N ) ..."
Pseudorandom Bit Generators That Fool Modular Sums
"... Abstract. We consider the following problem: for given n, M, produce a sequence X1,X2,...,Xn of bits that fools every linear test modulo M. We present two constructions of generators for such sequences. For every constant prime power M, the first construction has seed length OM (log(n/ɛ)), which is ..."
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Cited by 5 (1 self)
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Abstract. We consider the following problem: for given n, M, produce a sequence X1,X2,...,Xn of bits that fools every linear test modulo M. We present two constructions of generators for such sequences. For every constant prime power M, the first construction has seed length OM (log(n/ɛ)), which is optimal up to the hidden constant. (A similar construction was independently discovered by Meka and Zuckerman [MZ]). The second construction works for every M,n, and has seed length O(log n +log(M/ɛ)log(M log(1/ɛ))). The problem we study is a generalization of the problem of constructing small bias distributions [NN], which are solutions to the M =2case. We note that even for the case M = 3 the best previously known constructions were generators fooling general boundedspace computations, and required O(log 2 n) seed length. For our first construction, we show how to employ recently constructed generators for sequences of elements of ZM that fool smalldegree polynomials
EXPLICIT CONSTRUCTIONS OF RIP MATRICES AND RELATED PROBLEMS
"... Abstract. We give a new explicit construction of n × N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε> 0, large N and any n satisfying N 1−ε ≤ n ≤ N, we construct RIP matrices of order k 1/2+ε and constant δ −ε. This overcomes the natural barrier k = O(n 1/2) for proo ..."
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Cited by 4 (1 self)
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Abstract. We give a new explicit construction of n × N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε> 0, large N and any n satisfying N 1−ε ≤ n ≤ N, we construct RIP matrices of order k 1/2+ε and constant δ −ε. This overcomes the natural barrier k = O(n 1/2) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose kth moments are uniformly small for 1 ≤ k ≤ N (Turán’s power sum problem), which improves upon known explicit constructions when (log N) 1+o(1) ≤ n ≤ (log N) 4+o(1). This latter construction produces elementary explicit examples of n × N matrices that satisfy RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (log N) 1+o(1) ≤ n ≤ (log N) 5/2+o(1). 1.
Improved constructions of quantum automata
, 805
"... Abstract. We present a simple construction of quantum automata which achieve an exponential advantage over classical finite automata. Our automata use 4 log 2p + O(1) states to recognize a language that requires ǫ p states classically. The construction is both substantially simpler and achieves a be ..."
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Cited by 1 (0 self)
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Abstract. We present a simple construction of quantum automata which achieve an exponential advantage over classical finite automata. Our automata use 4 log 2p + O(1) states to recognize a language that requires ǫ p states classically. The construction is both substantially simpler and achieves a better constant in the front of log p than the previously known construction of [2]. Similarly to [2], our construction is by a probabilistic argument. We consider the possibility to derandomize it and present some results in this direction. 1
unknown title
"... The key to these constructions is a nearly optimal randomnessefficient version of the low degree test [32]. In a similar way we give a randomnessefficient version of the BLR linearity test [13] (which is used, for instance, in locally testing the Hadamard code). ..."
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The key to these constructions is a nearly optimal randomnessefficient version of the low degree test [32]. In a similar way we give a randomnessefficient version of the BLR linearity test [13] (which is used, for instance, in locally testing the Hadamard code).
Monotone expanders constructions and applications
"... The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. Constant degree ..."
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The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. Constant degree dimension expanders in finite fields, resolving a question of [BISW04]. 2. O(1)page and O(1)pushdown expanders, resolving a question of [GKS86], and leading to tight lower bounds on simulation time for certain Turing Machines. Bourgain [Bou09] gave recently an ingenious construction of such constant degree monotone expanders. The first application (1) above follows from a reduction in [DS08]. We give a short exposition of both construction and reduction. The new contributions of this paper are simple. First, we explain the observation leading to the second application (2) above, and some of its consequences. Second, we observe that a variant of the zigzag graph product preserves monotonicity, and use it to give a simple alternative construction of monotone expanders, with nearconstant degree. 1