The Shadow Theory of Modular and Unimodular Lattices

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 for 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}.