## The Shadow Theory of Modular and Unimodular Lattices

E. M. Rains and N. J. A. Sloane

Information Sciences Research, AT\&T Labs-Research

180 Park Avenue, Florham Park, NJ 07932-0971

Last revised May 20 1998

(This paper has now appeared in **Journal
of Number Theory**, Vol. **73** (1998), pp. 359-389.)

** 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
unimodualr 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 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 12-dimensional
Coxeter-Todd lattice. The even analogues of the Leech lattice
admit a uniform construction, inspired by the fact that (*) is the
set of square-free orders of elements of the Mathieu group M_23.

The full paper is available in
**postscript** or
**pdf** format.