The Experts below are selected from a list of 21342 Experts worldwide ranked by ideXlab platform
Emanuele Viterbo - One of the best experts on this subject based on the ideXlab platform.
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Algebraic Number Theory and code design for rayleigh fading channels
Foundations and Trends in Communications and Information Theory, 2004Co-Authors: Frederique Oggier, Emanuele ViterboAbstract:Algebraic Number Theory is having an increasing impact in code design for many different coding applications, such as single antenna fading channels and more recently, MIMO systems. Extended work has been done on single antenna fading channels, and Algebraic lattice codes have been proven to be an effective tool. The general framework has been settled in the last ten years and many explicit code constructions based on Algebraic Number Theory are now available.The aim of this work is to provide both an overview on Algebraic lattice code designs for Rayleigh fading channels, as well as a tutorial introduction to Algebraic Number Theory. The basic facts of this mathematical field will be illustrated by many examples and by the use of a computer algebra freeware in order to make it more accessible to a large audience.
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new Algebraic constructions of rotated z sup n lattice constellations for the rayleigh fading channel
IEEE Transactions on Information Theory, 2004Co-Authors: Eva Bayerfluckiger, Frederique Oggier, Emanuele ViterboAbstract:In this correspondence, we present various families of full diversity rotated Z/sup n/-lattice constellations based on Algebraic Number Theory constructions. We are able to give closed-form expressions of their minimum product distance using the corresponding Algebraic properties.
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good lattice constellations for both rayleigh fading and gaussian channels
IEEE Transactions on Information Theory, 1996Co-Authors: Joseph J Boutros, Emanuele Viterbo, C Rastello, Jeanclaude BelfioreAbstract:Recent work on lattices matched to the Rayleigh fading channel has shown how to construct good signal constellations with high spectral efficiency. We present a new family of lattice constellations, based on complex Algebraic Number fields, which have good performance on Rayleigh fading channels. Some of these lattices also present a reasonable packing density and thus may be used at the same time over a Gaussian channel. Conversely, we show that particular versions of the best lattice packings (D/sub 4/, E/sub 6/, E/sub 8/, K/sub 12/, /spl Lambda//sub 16/, /spl Lambda//sub 24/), constructed from totally complex Algebraic cyclotomic fields, present better performance over the Rayleigh fading channel. The practical interest in such signal constellations rises from the need to transmit information at high rates over both terrestrial and satellite links. Some further results in Algebraic Number Theory related to ideals and their factorization are presented and the decoding algorithm used with these lattice constellations are illustrated together with practical results.
Jacob L Bourjaily - One of the best experts on this subject based on the ideXlab platform.
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rooting out letters octagonal symbol alphabets and Algebraic Number Theory
Journal of High Energy Physics, 2020Co-Authors: Jacob L Bourjaily, Andrew J Mcleod, Cristian Vergu, Matthias Volk, Matt Von Hippel, Matthias WilhelmAbstract:It is widely expected that NMHV amplitudes in planar, maximally supersymmetric Yang-Mills Theory require symbol letters that are not rationally expressible in terms of momentum-twistor (or cluster) variables starting at two loops for eight particles. Re- cent advances in loop integration technology have made this an ‘experimentally testable’ hypothesis: compute the amplitude at some kinematic point, and see if Algebraic symbol letters arise. We demonstrate the feasibility of such a test by directly integrating the most difficult of the two-loop topologies required. This integral, together with its rotated image, suffices to determine the simplest NMHV component amplitude: the unique component finite at this order. Although each of these integrals involve Algebraic symbol alphabets, the combination contributing to this amplitude is — surprisingly — rational. We describe the steps involved in this analysis, which requires several novel tricks of loop integration and also a considerable degree of Algebraic Number Theory. We find dramatic and unusual simplifications, in which the two symbols initially expressed as almost ten million terms in over two thousand letters combine in a form that can be written in five thousand terms and twenty-five letters.
Frederique Oggier - One of the best experts on this subject based on the ideXlab platform.
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Algebraic Number Theory and code design for rayleigh fading channels
Foundations and Trends in Communications and Information Theory, 2004Co-Authors: Frederique Oggier, Emanuele ViterboAbstract:Algebraic Number Theory is having an increasing impact in code design for many different coding applications, such as single antenna fading channels and more recently, MIMO systems. Extended work has been done on single antenna fading channels, and Algebraic lattice codes have been proven to be an effective tool. The general framework has been settled in the last ten years and many explicit code constructions based on Algebraic Number Theory are now available.The aim of this work is to provide both an overview on Algebraic lattice code designs for Rayleigh fading channels, as well as a tutorial introduction to Algebraic Number Theory. The basic facts of this mathematical field will be illustrated by many examples and by the use of a computer algebra freeware in order to make it more accessible to a large audience.
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new Algebraic constructions of rotated z sup n lattice constellations for the rayleigh fading channel
IEEE Transactions on Information Theory, 2004Co-Authors: Eva Bayerfluckiger, Frederique Oggier, Emanuele ViterboAbstract:In this correspondence, we present various families of full diversity rotated Z/sup n/-lattice constellations based on Algebraic Number Theory constructions. We are able to give closed-form expressions of their minimum product distance using the corresponding Algebraic properties.
Fang Song - One of the best experts on this subject based on the ideXlab platform.
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efficient quantum algorithms for computing class groups and solving the principal ideal problem in arbitrary degree Number fields
Symposium on Discrete Algorithms, 2016Co-Authors: Jeanfrancois Biasse, Fang SongAbstract:This paper gives polynomial time quantum algorithms for computing the ideal class group (CGP) under the Generalized Riemann Hypothesis and solving the principal ideal problem (PIP) in Number fields of arbitrary degree. These are are fundamental problems in Number Theory and they are connected to many unproven conjectures in both analytic and Algebraic Number Theory. Previously the best known algorithms by Hallgren [20] only allowed to solve these problems in quantum polynomial time for Number fields of constant degree. In a recent breakthrough, Eisentrager et al. [11] showed how to compute the unit group in arbitrary fields, thus opening the way to the resolution of CGP and PIP in the general case. For example, Biasse and Song [3] pointed out how to directly apply this result to solve PIP in classes of cyclotomic fields of arbitrary degree.The methods we introduce in this paper run in quantum polynomial time in arbitrary classes of Number fields. They can be applied to solve other problems in computational Number Theory as well including computing the ray class group and solving relative norm equations. They are also useful for ongoing cryptanalysis of cryptographic schemes based on ideal lattices [5, 10].Our algorithms generalize the quantum algorithm for computing the (ordinary) unit group [11]. We first show that CGP and PIP reduce naturally to the computation of S-unit groups, which is another fundamental problem in Number Theory. Then we show an efficient quantum reduction from computing S-units to the continuous hidden subgroup problem introduced in [11]. This step is our main technical contribution, which involves careful analysis of the metrical properties of lattices to prove the correctness of the reduction. In addition, we show how to convert the output into an exact compact representation, which is convenient for further Algebraic manipulations.
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a quantum algorithm for computing the unit group of an arbitrary degree Number field
Symposium on the Theory of Computing, 2014Co-Authors: Kirsten Eisentrager, Sean Hallgren, Alexei Kitaev, Fang SongAbstract:Computing the group of units in a field of Algebraic Numbers is one of the central tasks of computational Algebraic Number Theory. It is believed to be hard classically, which is of interest for cryptography. In the quantum setting, efficient algorithms were previously known for fields of constant degree. We give a quantum algorithm that is polynomial in the degree of the field and the logarithm of its discriminant. This is achieved by combining three new results. The first is a classical algorithm for computing a basis for certain ideal lattices with doubly exponentially large generators. The second shows that a Gaussian-weighted superposition of lattice points, with an appropriate encoding, can be used to provide a unique representation of a real-valued lattice. The third is an extension of the hidden subgroup problem to continuous groups and a quantum algorithm for solving the HSP over the group Rn.
Jeanclaude Belfiore - One of the best experts on this subject based on the ideXlab platform.
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good lattice constellations for both rayleigh fading and gaussian channels
IEEE Transactions on Information Theory, 1996Co-Authors: Joseph J Boutros, Emanuele Viterbo, C Rastello, Jeanclaude BelfioreAbstract:Recent work on lattices matched to the Rayleigh fading channel has shown how to construct good signal constellations with high spectral efficiency. We present a new family of lattice constellations, based on complex Algebraic Number fields, which have good performance on Rayleigh fading channels. Some of these lattices also present a reasonable packing density and thus may be used at the same time over a Gaussian channel. Conversely, we show that particular versions of the best lattice packings (D/sub 4/, E/sub 6/, E/sub 8/, K/sub 12/, /spl Lambda//sub 16/, /spl Lambda//sub 24/), constructed from totally complex Algebraic cyclotomic fields, present better performance over the Rayleigh fading channel. The practical interest in such signal constellations rises from the need to transmit information at high rates over both terrestrial and satellite links. Some further results in Algebraic Number Theory related to ideals and their factorization are presented and the decoding algorithm used with these lattice constellations are illustrated together with practical results.