Delaunay polytopes derived from the Leech lattice

Mathieu Dutour Sikirić[1]; Konstantin Rybnikov[2]

  • [1] Rudjer Bosković Institute 54 ulica Bijenicka 1000 Zagreb, Croatia
  • [2] Department of Mathematical Sciences University of Massachusetts at Lowell MA 01854, Lowell, USA

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 1, page 85-101
  • ISSN: 1246-7405

Abstract

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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 ℝ 23 .

How to cite

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Dutour Sikirić, Mathieu, and Rybnikov, Konstantin. "Delaunay polytopes derived from the Leech lattice." Journal de Théorie des Nombres de Bordeaux 26.1 (2014): 85-101. <http://eudml.org/doc/275734>.

@article{DutourSikirić2014,
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 $\mathbb\{R\}^\{23\}$.},
affiliation = {Rudjer Bosković Institute 54 ulica Bijenicka 1000 Zagreb, Croatia; Department of Mathematical Sciences University of Massachusetts at Lowell MA 01854, Lowell, USA},
author = {Dutour Sikirić, Mathieu, Rybnikov, Konstantin},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Delaunay polytopes; Leech lattice; covering radius; lamination number},
language = {eng},
month = {4},
number = {1},
pages = {85-101},
publisher = {Société Arithmétique de Bordeaux},
title = {Delaunay polytopes derived from the Leech lattice},
url = {http://eudml.org/doc/275734},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Dutour Sikirić, Mathieu
AU - Rybnikov, Konstantin
TI - Delaunay polytopes derived from the Leech lattice
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/4//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 1
SP - 85
EP - 101
AB - 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 $\mathbb{R}^{23}$.
LA - eng
KW - Delaunay polytopes; Leech lattice; covering radius; lamination number
UR - http://eudml.org/doc/275734
ER -

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