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Mathieu Dutour Sikirić; Konstantin Rybnikov
Delaunay polytopes derived from the Leech lattice
Journal de théorie des nombres de Bordeaux, 26 no. 1 (2014), p. 85-101, doi: 10.5802/jtnb.860
Article PDF | Reviews MR 3232768 | Zbl 06304182
Class. Math.: 11H31, 11H55

Résumé - Abstract

A Delaunay polytope in a lattice $L$ is perfect if any affine transformation that preserve its Delaunay property is a composite of an homothety and an isometry. Perfect Delaunay polytopes are rare in low dimension and here we consider the ones that one can get in lattice that are sections of the Leech lattice.

By doing so we are able to find lattices with several orbits of perfect Delaunay polytopes. Also we exhibit Delaunay polytopes which remain Delaunay in some superlattices. We found perfect Delaunay polytopes with small automorphism group relative to the automorphism group of the lattice. And we prove that some perfect Delaunay polytopes have lamination number $5$, which is higher than previously known $3$.

A well known construction of centrally symmetric perfect Delaunay polytopes uses a laminated construction from an antisymmetric perfect Delaunay polytope. We fully classify the types of perfect Delaunay polytopes that can occur.

Finally, we derived an upper bound for the covering radius of $\Lambda _{24}(v)^{*}$, which generalizes the Smith bound and we prove that this bound is met only by $\Lambda _{23}^{*}$, the best known lattice covering in $\mathbb{R}^{23}$.


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