Automorphisms of extremal unimodular lattices in dimension 72

Nebe, Gabriele

doi:10.1016/j.jnt.2015.05.001

Cited by 6 publications

(3 citation statements)

References 38 publications

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“…One might of course try to improve by considering other lattices and their orbifolds. However, [Neb13] suggests that other extremal lattices, if they exist, have smaller automorphism groups; [Neb16] gives a similar result for Γ 72 . Using non-extremal lattices on the other hand introduces more light vectors coming from the short vectors of the lattice, which need to be eliminated by a powerful enough orbifold group.…”

Section: Introductionmentioning

confidence: 82%

Orbifolds of lattice vertex operator algebras at d = 48 and d = 72

Gemünden¹,

Keller²

2019

Journal of Algebra

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We construct orbifolds of holomorphic lattice Vertex Operator Algebras for non-Abelian finite automorphism groups G. To this end, we construct twisted modules for automorphisms g together with the projective representation of the centralizer of g on the twisted module. This allows us to extract the irreducible modules of the fixed point VOA V G , and to compute their characters and modular transformation properties. We then construct holomorphic VOAs by adjoining such modules to V G . Applying these methods to extremal lattices in d = 48 and d = 72, we construct more than fifty new holomorphic VOAs of central charge 48 and 72, many of which have a very small number of light states. Twisted Modules, Orbifolds and extensions

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Section: Introductionmentioning

confidence: 82%

Orbifolds of lattice vertex operator algebras at d = 48 and d = 72

Gemünden¹,

Keller²

2019

Journal of Algebra

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“…d−modular lattices which reach those bounds are called extremal. Classification of known extremal lattices is found in [1,9,23,24,26,32] and on the on-line table [36].…”

Section: Interesting Lattices From Totally Real Quadratic Fieldsmentioning

confidence: 99%

Modular lattices from a variation of construction a over number fields

Hou¹,

Oggier²

2017

Advances in Mathematics of Communications

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We consider a variation of Construction A of lattices from linear codes based on two classes of number fields, totally real and CM Galois number fields. We propose a generic construction with explicit generator and Gram matrices, then focus on modular and unimodular lattices, obtained in the particular cases of totally real, respectively, imaginary, quadratic fields. Our motivation comes from coding theory, thus some relevant properties of modular lattices, such as minimal norm, theta series, kissing number and secrecy gain are analyzed. Interesting lattices are exhibited.

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“…Currently, the exact number of such lattices is still an open problem in higher dimensions. For example, we only know of a single one in dimension 72 [Neb16] and four in dimension 80.…”

Section: Introductionmentioning

confidence: 99%

Even Unimodular Lattices from Quaternion Algebras

Amorós,

Damir,

Hollanti

2021

Preprint

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We review a lattice construction arising from quaternion algebras over number fields and use it to obtain some known extremal and densest lattices in dimensions 8 and 16. The benefit of using quaternion algebras over number fields is that the dimensionality of the construction problem is reduced by 3/4. We explicitly construct the E8 lattice (resp. E 2 8 and Λ16) from infinitely many quaternion algebras over real quadratic (resp. quartic) number fields and we further present a density result on such number fields. By relaxing the extremality condition, we also provide a source for constructing even unimodular lattices in any dimension multiple of 8.

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Automorphisms of extremal unimodular lattices in dimension 72

Cited by 6 publications

References 38 publications

Orbifolds of lattice vertex operator algebras at d = 48 and d = 72

Orbifolds of lattice vertex operator algebras at d = 48 and d = 72

Modular lattices from a variation of construction a over number fields

Even Unimodular Lattices from Quaternion Algebras

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Automorphisms of extremal unimodular lattices in dimension 72

Cited by 6 publications

References 38 publications

Orbifolds of lattice vertex operator algebras at d = 48 and d = 72

Orbifolds of lattice vertex operator algebras at d = 48 and d = 72

Modular lattices from a variation of construction a over number fields

Even Unimodular Lattices from Quaternion Algebras

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Product

Resources

About

Orbifolds of lattice vertex operator algebras at d = 48 and d = 72

Orbifolds of lattice vertex operator algebras at d = 48 and d = 72