S-extremal strongly modular lattices

Gabriele Nebe[1]; Kristina Schindelar[1]

  • [1] Lehrstuhl D für Mathematik RWTH Aachen 52056 Aachen, Germany

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 3, page 683-701
  • ISSN: 1246-7405

Abstract

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S-extremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the s-extremal unimodular lattices defined in [6]. If the minimum of the lattice is even, then the dimension of an s-extremal lattices can be bounded by the theory of modular forms. This shows that such lattices are also extremal and that there are only finitely many s-extremal strongly modular lattices of even minimum.

How to cite

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Nebe, Gabriele, and Schindelar, Kristina. "S-extremal strongly modular lattices." Journal de Théorie des Nombres de Bordeaux 19.3 (2007): 683-701. <http://eudml.org/doc/249978>.

@article{Nebe2007,
abstract = {S-extremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the s-extremal unimodular lattices defined in [6]. If the minimum of the lattice is even, then the dimension of an s-extremal lattices can be bounded by the theory of modular forms. This shows that such lattices are also extremal and that there are only finitely many s-extremal strongly modular lattices of even minimum.},
affiliation = {Lehrstuhl D für Mathematik RWTH Aachen 52056 Aachen, Germany; Lehrstuhl D für Mathematik RWTH Aachen 52056 Aachen, Germany},
author = {Nebe, Gabriele, Schindelar, Kristina},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {extremal strongly modular lattices},
language = {eng},
number = {3},
pages = {683-701},
publisher = {Université Bordeaux 1},
title = {S-extremal strongly modular lattices},
url = {http://eudml.org/doc/249978},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Nebe, Gabriele
AU - Schindelar, Kristina
TI - S-extremal strongly modular lattices
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 3
SP - 683
EP - 701
AB - S-extremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the s-extremal unimodular lattices defined in [6]. If the minimum of the lattice is even, then the dimension of an s-extremal lattices can be bounded by the theory of modular forms. This shows that such lattices are also extremal and that there are only finitely many s-extremal strongly modular lattices of even minimum.
LA - eng
KW - extremal strongly modular lattices
UR - http://eudml.org/doc/249978
ER -

References

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  11. H.-G. Quebbemann, Atkin-Lehner eigenforms and strongly modular lattices. L’Ens. Math. 43 (1997), 55–65. Zbl0898.11014
  12. E.M. Rains, New asymptotic bounds for self-dual codes and lattices. IEEE Trans. Inform. Theory 49 (2003), no. 5, 1261–1274. Zbl1063.94123MR1984825
  13. E.M. Rains, N.J.A. Sloane, The shadow theory of modular and unimodular lattices. J. Number Th. 73 (1998), 359–389. Zbl0917.11026MR1657980
  14. R. Scharlau, R. Schulze-Pillot, Extremal lattices. In Algorithmic algebra and number theory, Herausgegeben von B. H. Matzat, G. M. Greuel, G. Hiss. Springer, 1999, 139–170. Zbl0944.11012MR1672117
  15. K. Schindelar, Stark modulare Gitter mit langem Schatten. Diplomarbeit, Lehrstuhl D für Mathematik, RWTH Aachen (2006). 
  16. E.T. Whittaker, G.N. Watson, A course of modern analysis (4th edition) Cambridge University Press, 1963. Zbl0108.26903MR1424469

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