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A generalization of some lattices of Coxeter. (English) Zbl 1106.11026

For each dimension \(n\geq 10\) there is an integral perfect lattice of odd minimum with the additional property that all proper sections of the same minimum are not perfect. The construction combines ideas from Watson and Coxeter and is based on the geometry of a specific set of \(n(n+1)/2\) minimal vectors.

MSC:

11H55 Quadratic forms (reduction theory, extreme forms, etc.)
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References:

[1] Watson, Mathematika. 18 pp 60– (1971)
[2] Martinet, J. Algebra and Analysis (Saint-Petersburg) 16 pp 198– (2004)
[3] Martinet, Reseaux Euclidiens, Designs Sphériques et Groupes pp 163– (2001)
[4] Bacher, Réseaux Euclidiens, Designs Spheriques et Groupes 37 pp 212– (2001)
[5] Coxeter, Canad. J. Math. 3 pp 391– (1951) · Zbl 0044.04201 · doi:10.4153/CJM-1951-045-8
[6] DOI: 10.1023/B:JACO.0000047289.61038.1f · Zbl 1054.05093 · doi:10.1023/B:JACO.0000047289.61038.1f
[7] Martinet, Perfect Lattices in Euclidean Spaces (2003) · doi:10.1007/978-3-662-05167-2
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