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Deterministic Extractors for Affine Sources over Large Fields
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 108 (2005)
, 2005
"... An (n, k)affine source over a finite field F is a random variable X = (X1,..., Xn) ∈ Fn, which is uniformly distributed over an (unknown) kdimensional affine subspace of F n. We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size large ..."
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Cited by 38 (6 self)
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An (n, k)affine source over a finite field F is a random variable X = (X1,..., Xn) ∈ Fn, which is uniformly distributed over an (unknown) kdimensional affine subspace of F n. We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size larger than n c (where c is a large enough constant). Our main results are as follows: 1. (For arbitrary k): For any n, k and any F of size larger than n 20, we give an explicit construction for a function D: F n → F k−1, such that for any (n, k)affine source X over F, the distribution of D(X) is ɛclose to uniform, where ɛ is polynomially small in F. 2. (For k = 1): For any n and any F of size larger than n c, we give an explicit construction for a function D: F n → {0, 1} (1−δ) log 2 F  , such that for any (n, 1)affine source X over F, the distribution of D(X) is ɛclose to uniform, where ɛ is polynomially small in F. Here, δ> 0 is an arbitrary small constant, and c is a constant depending on δ.
Erdős distance problem in vector spaces over finite fields
 Transactions of the American Mathematical Society
"... Abstract. We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let Fq be a finite field with q elements and take E ⊂ F d q, d ≥ 2. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in F ..."
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Cited by 36 (10 self)
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Abstract. We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let Fq be a finite field with q elements and take E ⊂ F d q, d ≥ 2. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in F d q to provide estimates for minimum cardinality of the distance set ∆(E) intermsofthe cardinality of E. Bounds for Gauss and Kloosterman sums play an important role in the proof. 1.
On QuasiOrthogonal Signatures for CDMA Systems
 IEEE Trans. Inform. Theory. [Online]. Available: http://www.ece.utexas.edu/˜rheath/papers/2002/quasicdma
, 2002
"... Codebooks constructed from Welch bound equality sequences are optimal in terms of sum capacity in synchronous CDMA systems. These optimal codebooks depend on both the sequence length as well as the number of active sequences. Thus to maintain optimality a typically different codebook is needed fo ..."
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Cited by 22 (7 self)
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Codebooks constructed from Welch bound equality sequences are optimal in terms of sum capacity in synchronous CDMA systems. These optimal codebooks depend on both the sequence length as well as the number of active sequences. Thus to maintain optimality a typically different codebook is needed for every possible number of active users. Further, all the sequences need to be reassigned as the number of active users changes. This paper describes and analyzes two promising subclasses of Welch bound equality sequences that have good properties when only a subset of sequences is active. One subclass, based on maximum Welch bound equality sequence sets, has equiangular sequences thus each user experiences the same amount of interference. The interference power depends only on the total number of active users. Another subclass, constructed by concatenating multiple orthonormal basis, comes closer to the Welch bound when not all signatures are active. Optimal unions of orthonormal basis are derived. Performance when subsets of sequences are active is characterized in terms of sumsquared correlation, average interference, and condition number of the Gram matrix.
Sumproduct Estimates in Finite Fields via Kloosterman Sums
"... We establish improved sumproduct bounds in finite fields using incidence theorems based on bounds for classical Kloosterman and related sums. ..."
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Cited by 19 (2 self)
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We establish improved sumproduct bounds in finite fields using incidence theorems based on bounds for classical Kloosterman and related sums.
Quantum algorithms for hidden nonlinear structures
 Proc. 48th IEEE Symp. on Found. Comp. Sci., IEEE
"... Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor’s factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden n ..."
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Cited by 17 (4 self)
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Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor’s factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures. 1
The analytic rank of J0(q) and zeros of automorphic Lfunctions
"... We study zeros of the Lfunctions L(f, s) of primitive weight two forms of level q. Our main result is that, on average over forms f of level q prime, the order of the Lfunctions at the central critical point s = 1 2 is absolutely bounded. On the Birch and SwinnertonDyer conjecture, this provides ..."
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Cited by 17 (11 self)
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We study zeros of the Lfunctions L(f, s) of primitive weight two forms of level q. Our main result is that, on average over forms f of level q prime, the order of the Lfunctions at the central critical point s = 1 2 is absolutely bounded. On the Birch and SwinnertonDyer conjecture, this provides an upper bound for the rank of the Jacobian J0(q) of the modular curve X0(q), which is of the same order of magnitude as what is expected to be true. 1
The distribution of the free path lengths in the periodic twodimensional Lorentz gas in the smallscatterer limit
, 2003
"... We study the free path length and the geometric free path length in the model of the periodic twodimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the smallscatterer limit and explicitly compute them. As a corollary one get ..."
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Cited by 17 (6 self)
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We study the free path length and the geometric free path length in the model of the periodic twodimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the smallscatterer limit and explicitly compute them. As a corollary one gets a complete proof for the existence of the constant term c = 2 − 3ln 2 + 27ζ(3) 2π2 in the asymptotic formula h(T) = −2ln ε + c + o(1) of the KS entropy of the billiard map in this model.
ARITHMETIC QUANTUM UNIQUE ERGODICITY FOR SYMPLECTIC LINEAR MAPS OF THE MULTIDIMENSIONAL TORUS
, 2006
"... We look at the expectation values for quantized linear symplectic maps on the multidimensional torus and their distribution in the semiclassical limit. We construct superscars that are stable under the arithmetic symmetries of the system and localize around invariant manifolds. We show that these ..."
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Cited by 17 (3 self)
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We look at the expectation values for quantized linear symplectic maps on the multidimensional torus and their distribution in the semiclassical limit. We construct superscars that are stable under the arithmetic symmetries of the system and localize around invariant manifolds. We show that these superscars exist only when there are isotropic rational subspaces, invariant under the linear map. In the case where there are no such scars, we compute the variance of the fluctuations of the matrix elements for the desymmetrized system, and present a conjecture for their limiting distributions.