The shadow theory of modular and unimodular lattices

[摘要] It is shown that an n-dimensional unimodular lattice has minimal norm at most 2[n/24] + 2, unless n = 23 when the bound must be increased by 1. This result was previously known only for even unimodular lattices. Quebbemann had extended the bound fur even unimodular lattices to strongly N-modular even lattices for N in (*) {1, 2, 3, 5, 6, 7, 11, 14, 15, 23}, and analogous bounds are established here for odd lattices satisfying certain technical conditions (which are trivial for N=1 and 2). For N>1 in (*), lattices meeting the new bound are constructed that are analogous to the shorter and odd Leech lattices. These include an odd associate of the 16-dimensional Barnes-Wall lattice and shorter and odd associates of the Coxeter-Todd lattice. A uniform construction is given For the (even) analogues of the Leech lattice, inspired by the fact that (*) is also the set of square-free orders of elements of the Mathieu group M-23. (C) 1998 Academic Press.

[发布日期] 1998-12-01