The genus of the Barnes-Wall lattice
type: Item
instance of: scholarly article
title: The genus of the Barnes-Wall lattice
zbMATH Open document ID: 0815.11034
DOI: 10.1007/BF02564490
author: Rudolf Scharlau
author: B. Venkov
published in: Commentarii Mathematici Helvetici
publication date: 1995-07-09T00:00:00Z
full work available at URL: https://eudml.org/doc/140342
review text: In analogy with the well-known case of even unimodular lattices in the Euclidean space of dimension 24 (the genus of the Leech lattice), the present paper gives the classification of the lattices in the genus of the 16-dimensional Barnes-Wall lattice. The latter has already been known to be the unique lattice of minimum norm 4 in its genus, and the authors find that, again, there are exactly 23 other ones with minimum norm 2, each corresponding to a certain root system of rank 16. Some further results of classifications in lower dimensions are only mentioned. In particular, for the odd primes \(p\) such that \(p + 1\) divides 24, it is confirmed that in dimension \(48/(p + 1)\) there always exists a unique ``extremal'' even lattice isometric to its \(p\)-scaled dual (as proposed by the reviewer), and all other lattices in their genera have been found.
Mathematics Subject Classification ID: 11H55
Mathematics Subject Classification ID: 11E41
zbMATH DE Number: 720100
zbMATH Keywords: reflective lattice
zbMATH Keywords: extremal lattice
zbMATH Keywords: Barnes-Wall lattice
reviewed by: Heinz-Georg Quebbemann
MaRDI profile type: Publication
Wikidata QID: Q29031318
description: scientific article; zbMATH DE number 720100