论文信息 - S-extremal strongly modular lattices

S-extremal strongly modular lattices

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 dened 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 nitely many s-extremal strongly modular lattices of even minimum.

Gabriele Nebe | Kristina Schindelar | G. Nebe | Kristina Schindelar

[1] Noam D. Elkies. Lattices and codes with long shadows , 1995 .

[2] Thomas C. Hales. Sphere packings, I , 1997, Discret. Comput. Geom..

[3] R. Guy. Review: J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups , 1989 .

[4] P. Gaborit. A bound for certain s-extremal lattices and codes , 2007 .

[5] Eric M. Rains. New asymptotic bounds for self-dual codes and lattices , 2003, IEEE Trans. Inf. Theory.

[6] J. H. Conway,et al. A Note on Optimal Unimodular Lattices , 1998 .

[7] Neil J. A. Sloane,et al. Upper bounds for modular forms, lattices, and codes , 1975 .

[8] Edmund Taylor Whittaker,et al. A Course of Modern Analysis , 2021 .

[9] Rudolf Scharlau,et al. Extremal Lattices , 1997, Algorithmic Algebra and Number Theory.

[10] M. Kneser,et al. Klassenzahlen definiter quadratischer Formen , 1957 .

[11] Helmut Hasse,et al. Number Theory , 2020, An Introduction to Probabilistic Number Theory.

[12] W. Fischer,et al. Sphere Packings, Lattices and Groups , 1990 .

[13] Gabriele Nebe,et al. Unimodular lattices with long shadow , 2003 .

[14] N. Sloane,et al. The Shadow Theory of Modular and Unimodular Lattices , 1998, math/0207294.

[15] Strongly modular lattices with long shadow , 2002, math/0211232.