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30
On the decoding of algebraicgeometric codes
 IEEE TRANS. INFORM. THEORY
, 1995
"... This paper provides a survey of the existing literature on the decoding of algebraicgeometric codes. Definitions, theorems and cross references will be given. We show what has been done, discuss what still has to be done and pose some open problems. The following subjects are examined in a more or ..."
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Cited by 25 (5 self)
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This paper provides a survey of the existing literature on the decoding of algebraicgeometric codes. Definitions, theorems and cross references will be given. We show what has been done, discuss what still has to be done and pose some open problems. The following subjects are examined in a more or less historical order.
Computing RiemannRoch spaces in algebraic function fields and related topics
, 2001
"... this paper we develop a simple and efficient algorithm for the computation of RiemannRoch spaces to be counted among the arithmetic methods. The algorithm completely avoids series expansions and resulting complications, and instead relies on integral closures and their ideals only. It works for any ..."
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Cited by 22 (0 self)
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this paper we develop a simple and efficient algorithm for the computation of RiemannRoch spaces to be counted among the arithmetic methods. The algorithm completely avoids series expansions and resulting complications, and instead relies on integral closures and their ideals only. It works for any "computable" constant field k of any characteristic as long as the required integral closures can be computed, and does not involve constant field extensions
On a refined Stark Conjecture for function fields
 COMP. MATH
, 1996
"... We prove that a refinement of Stark's Conjecture formulated by Rubin in [14] is true up to primes dividing the order of the Galois group, for finite, abelian extensions of function fields over finite fields. We also show that in the case of constant field extensions a statement stronger than Ru ..."
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Cited by 16 (7 self)
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We prove that a refinement of Stark's Conjecture formulated by Rubin in [14] is true up to primes dividing the order of the Galois group, for finite, abelian extensions of function fields over finite fields. We also show that in the case of constant field extensions a statement stronger than Rubin's holds true.
Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are ..."
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Cited by 16 (3 self)
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1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1
Sato–Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height
, 2008
"... ..."
Decoding Codes from Curves and Cyclic Codes
, 1993
"... 4. with R. K"otter, "Errorlocating pairs for cyclic codes, " preprint EindhovenLink"oping, submitted for publication, March 1993. ..."
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Cited by 9 (0 self)
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4. with R. K&quot;otter, &quot;Errorlocating pairs for cyclic codes, &quot; preprint EindhovenLink&quot;oping, submitted for publication, March 1993.
Grastype Conjectures for Function Fields
 Comp. Math
"... Based on results obtained in [15], we construct groups of special S units for function fields of characteristic p > 0, and show that they satisfy Gras type Conjectures. We use these results in order to give a new proof of Chinburg's # 3 Conjecture on the Galois module structure of th ..."
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Cited by 8 (5 self)
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Based on results obtained in [15], we construct groups of special S units for function fields of characteristic p > 0, and show that they satisfy Gras type Conjectures. We use these results in order to give a new proof of Chinburg's # 3 Conjecture on the Galois module structure of the group of Sunits, for cyclic extensions of prime degree of function fields. 0.
Applications of Exponential Sums in Communications Theory
, 1999
"... We provide an introductory overview of how exponential sums, and bounds for them, have been exploited by coding theorists and communications engineers. ..."
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Cited by 5 (0 self)
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We provide an introductory overview of how exponential sums, and bounds for them, have been exploited by coding theorists and communications engineers.
Cyclicity of CM elliptic curves modulo p
 TRANSACTIONS OF AMERICAN MATHEMATICAL SOCIETY
, 2003
"... Let E be an elliptic curve defined over Q and with complex multiplication. For a prime p of good reduction, let E be the reduction of E modulo p. We find the density of the primes p ≤ x for which E(Fp) is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.P. ..."
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Cited by 4 (1 self)
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Let E be an elliptic curve defined over Q and with complex multiplication. For a prime p of good reduction, let E be the reduction of E modulo p. We find the density of the primes p ≤ x for which E(Fp) is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new simpler unconditional proof of this asymptotic formula, and also to provide explicit error terms in the formula.