The Shadow Theory of Modular and Unimodular Lattices
Abstract
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}.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- July 2002
- DOI:
- 10.48550/arXiv.math/0207294
- arXiv:
- arXiv:math/0207294
- Bibcode:
- 2002math......7294R
- Keywords:
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- Combinatorics;
- 11H31 (11H50;
- 11H56)
- E-Print:
- 33 pages