Delaunay polytopes derived from the Leech lattice
Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 1, pp. 85-101.

Un polytope de Delaunay dans un réseau L est parfait si les transformations affine qui préservent sa propriété d’être Delaunay sont des compositions d’homothéties et d’isométries. Les polytopes de Delaunay parfaits sont rares en petite dimension et ici nous considérons ceux qui apparaissent dans des sections du réseau de Leech.

Par ce moyen, nous trouvons des rśeaux ayant plusieurs orbites de polytopes de Delaunay parfaits. Nous exhibons aussi des polytopes de Delaunay parfait qui restent Delaunay dans des super-réseaux. Aussi nous trouvons des polytopes de Delaunay parfait ayant des groupes d’automorphismes relativement petit par rapport à leurs réseaux. Nous prouvons aussi que certains polytopes de Delaunay parfaits ont un nombre de lamination égal à 5 alors que les polytopes de Delaunay précédemment connus ont un nombre de lamination égal à 3.

Une construction bien connu de polytopes de Delaunay centralement symmétriques utilise des polytope de Delaunay parfait antisymmétriques. Nous classifions complètement les types de polytopes de Delaunay parfaits qui peuvent apparaîtrent dans cette construction.

Enfin, nous prouvons une borne supérieure sur le rayon de recouvrement du réseau Λ 24 (v) * qui généralise la borne de Smith. Nous prouvons que cette borne est atteinte seulement pour Λ 23 * qui est le meilleur recouvrement de 23 connu.

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 Λ 24 (v) * , which generalizes the Smith bound and we prove that this bound is met only by Λ 23 * , the best known lattice covering in 23 .

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DOI : https://doi.org/10.5802/jtnb.860
Classification : 11H31,  11H55
@article{JTNB_2014__26_1_85_0,
     author = {Mathieu Dutour Sikiri\'c and Konstantin Rybnikov},
     title = {Delaunay polytopes derived from  the {Leech} lattice},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {85--101},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {1},
     year = {2014},
     doi = {10.5802/jtnb.860},
     mrnumber = {3232768},
     zbl = {1317.11075},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.860/}
}
Mathieu Dutour Sikirić; Konstantin Rybnikov. Delaunay polytopes derived from  the Leech lattice. Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 1, pp. 85-101. doi : 10.5802/jtnb.860. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.860/

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