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.)


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.

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