The genus of the Barnes-Wall lattice
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.
scientific article; zbMATH DE number 720100