  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}.
