The Shadow Theory of Modular and Unimodular Lattices

✍ Scribed by E.M Rains; N.J.A Sloane

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 452 KB
• Volume: 73
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1998.2306

✦ Synopsis

It is shown that an n-dimensional unimodular lattice has minimal norm at most 2[nÂ24]+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],

( V)

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 (V), 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 (V) is also the set of square-free orders of elements of the Mathieu group M 23 .