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26
Construction of asymptotically good, lowrate errorcorrecting codes through pseudorandom graphs
 IEEE Transactions on Information Theory
, 1992
"... A new technique, based on the pseudorandom properties of certain graphs, known as expanders, is used to obtain new simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling and then regrouping ..."
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Cited by 117 (24 self)
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A new technique, based on the pseudorandom properties of certain graphs, known as expanders, is used to obtain new simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling and then regrouping the code coordinates. For any fixed (small) rate, and for sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF (2)) as well. Although these concatenated codes lie below Zyablov bound, they are still superior to previouslyknown explicit constructions in the zerorate neighborhood.
Explicit construction of linear sized tolerant networks
, 2006
"... For every ɛ > 0 and every integer m > 0, we construct explicitly graphs with O(m/ɛ) vertices and maximum degree O(1/ɛ²), such that after removing any (1 − ɛ) portion of their vertices or edges, the remaining graph still contains a path of length m. This settles a problem of Rosenberg, which was moti ..."
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Cited by 107 (14 self)
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For every ɛ > 0 and every integer m > 0, we construct explicitly graphs with O(m/ɛ) vertices and maximum degree O(1/ɛ²), such that after removing any (1 − ɛ) portion of their vertices or edges, the remaining graph still contains a path of length m. This settles a problem of Rosenberg, which was motivated by the study of fault tolerant linear arrays.
Primality testing with Gaussian periods
, 2003
"... The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new ..."
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Cited by 18 (0 self)
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The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the
On Computing Factors of Cyclotomic Polynomials
, 1993
"... For odd squarefree n > 1 the cyclotomic polynomial n (x) satises the identity of Gauss 4 n (x) = A 2 n ( 1) (n 1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is n (( 1) (n 1)=2 x) = C 2 n nxD 2 n or, in the case that n is even and squarefree, n=2 ( x 2 ) ..."
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Cited by 14 (5 self)
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For odd squarefree n > 1 the cyclotomic polynomial n (x) satises the identity of Gauss 4 n (x) = A 2 n ( 1) (n 1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is n (( 1) (n 1)=2 x) = C 2 n nxD 2 n or, in the case that n is even and squarefree, n=2 ( x 2 ) = C 2 n nxD 2 n ; Here A n (x); : : : ; D n (x) are polynomials with integer coecients. We show how these coef cients can be computed by simple algorithms which require O(n 2 ) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for A n (x); : : : ; D n (x), and illustrate the application to integer factorization with some numerical examples.
The Riemann hypothesis for certain integrals of Eisenstein series
 J. Number Theory
"... Abstract. This paper studies the nonholomorphic Eisenstein series E(z, s) for the modular surface PSL(2, Z)\H, and shows that integration with respect to certain nonnegative measures µ(z) gives meromorphic functions Fµ(s) that have all their zeros on the line ℜ(s) = 1 2. For the constant term a0( ..."
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Cited by 8 (2 self)
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Abstract. This paper studies the nonholomorphic Eisenstein series E(z, s) for the modular surface PSL(2, Z)\H, and shows that integration with respect to certain nonnegative measures µ(z) gives meromorphic functions Fµ(s) that have all their zeros on the line ℜ(s) = 1 2. For the constant term a0(y, s) of the Eisenstein series the Riemann hypothesis holds for all values y ≥ 1, with at most two exceptional real zeros, which occur exactly for those y> 4πe −γ = 7.0555+. The Riemann hypothesis holds for all truncation integrals with truncation parameter T ≥ 1. At the value T = 1 this proves the Riemann hypothesis for a zeta function Z2,Q(s) recently introduced by Lin Weng, associated to rank 2 semistable lattices over Q. 1.
Yıldırım, Small gaps between primes or almost primes
"... Abstract. Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that (pn+1 − pn) lim inf =0. n→ ∞ log pn We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of ex ..."
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Cited by 8 (2 self)
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Abstract. Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that (pn+1 − pn) lim inf =0. n→ ∞ log pn We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of exactly two distinct primes. We prove that lim inf n→ ∞ (qn+1 − qn) ≤ 26. If an appropriate generalization of the ElliottHalberstam Conjecture is true, then the above bound can be improved to 6. 1.
LocationCorrecting Codes
 IEEE Trans. Inform. Theory
, 1997
"... We study codes over GF (q) that can correct t channel errors assuming the error values are known. This is a counterpart to the wellknown problem of erasure correction, where error values are found assuming the locations are known. The correction capabilities of these so called tlocation correcting ..."
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Cited by 7 (1 self)
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We study codes over GF (q) that can correct t channel errors assuming the error values are known. This is a counterpart to the wellknown problem of erasure correction, where error values are found assuming the locations are known. The correction capabilities of these so called tlocation correcting codes (tLCCs) are characterized by a new metric, the decomposability distance, which plays a role analogous to that of the Hamming metric in conventional errorcorrecting codes (ECCs). Based on the new metric, we present bounds on the parameters of t LCCs that are counterparts to the classical Singleton, sphere packing and GilbertVarshamov bounds for ECCs. In particular, we show examples of perfect LCCs, and we study optimal (MDSlike) LCCs that attain the Singletontype bound on the redundancy. We show that these optimal codes are generally much shorter than their erasure (or conventional ECC) analogues: The length n of any tLCC that attains the Singletontype bound for t ? 1 is bounde...
On a twovariable zeta function for number fields
 Annales Inst. Fourier
"... This paper studies a twovariable zeta function ZK(w,s) attached to an algebraic number field K, introduced by van der Geer and Schoof [9], which is based on an analogue of the RiemannRoch theorem for number fields using Arakelov divisors. We term it the Arakelov zeta function. When w = 1 this func ..."
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Cited by 6 (0 self)
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This paper studies a twovariable zeta function ZK(w,s) attached to an algebraic number field K, introduced by van der Geer and Schoof [9], which is based on an analogue of the RiemannRoch theorem for number fields using Arakelov divisors. We term it the Arakelov zeta function. When w = 1 this function becomes the completed Dedekind zeta ˆ ζK(s) function of the field K. The function is an meromorphic function of two complex variables with polar divisor s(w − s), and it satisfies the functional equation ZK(w,s) = ZK(w,w − s). We consider the special case K = Q, where for w = 1 this function is ˆ s ζ(s) = π 2Γ ( s
Unextendible product bases
 J. Combinatorial Theory, Ser. A
"... Let C denote the complex field. A vector v in the tensor product ⊗ m i=1 Cki is called a pure product vector if it is a vector of the form v1 ⊗ v2 · · · ⊗ vm, with vi ∈ C ki. A set F of pure product vectors is called an unextendible product basis if F consists of orthogonal nonzero vectors, and t ..."
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Cited by 6 (0 self)
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Let C denote the complex field. A vector v in the tensor product ⊗ m i=1 Cki is called a pure product vector if it is a vector of the form v1 ⊗ v2 · · · ⊗ vm, with vi ∈ C ki. A set F of pure product vectors is called an unextendible product basis if F consists of orthogonal nonzero vectors, and there is no nonzero pure product vector in ⊗ m i=1 Cki which is orthogonal to all members of F. The construction of such sets of small cardinality is motivated by a problem in quantum information theory. Here it is shown that the minimum possible cardinality of such a set F is precisely 1 + � m i=1 (ki − 1) for every sequence of integers k1, k2,..., km ≥ 2 unless either (i) m = 2 and 2 ∈ {k1, k2} or (ii)1+ � m i=1 (ki −1) is odd and at least one ki is even. In each of these two cases, the minimum cardinality of the corresponding F is strictly bigger than 1 + � m i=1 (ki − 1). 1