The Shadow Theory of Modular and Unimodular Lattices
Abstract
It is shown that an ndimensional 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 Nmodular 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 16dimensional BarnesWall lattice and shorter and odd associates of the CoxeterTodd lattice. A uniform construction is given for the (even) analogues of the Leech lattice, inspired by the fact that (*) is also the set of squarefree orders of elements of the Mathieu group M_{23}.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 July 2002
 arXiv:
 arXiv:math/0207294
 Bibcode:
 2002math......7294R
 Keywords:

 Combinatorics;
 11H31 (11H50;
 11H56)
 EPrint:
 33 pages