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).

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