The Shadow Theory of Modular and Unimodular Lattices
The Shadow Theory of Modular and Unimodular Lattices
复制标题
模和幺模格子的影子理论
DOI：
10.1006/jnth.1998.2306
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发表时间：
1998
影响因子：
0.7
通讯作者：
N. Sloane
中科院分区：
文献类型：
--
作者：
E. Rains;N. Sloane
本文证明了n维么模格的最小范数至多为2[ n /24]+2，除非n =23，否则当界必须增加1时。这个结果以前只知道偶幺模格。Quebbemann已经扩大了界限，甚至unimodular格强N -模偶格为N在{1，2，3，5，6，7，11，14，15，23}，（*）和类似的界限建立在这里的奇格满足一定的技术条件（这是平凡的N =1和2）。当N >1时，满足新界的格被构造成类似于“短”和“奇”Leech格。这些包括一个奇怪的16维巴恩斯-沃尔晶格和更短的和奇怪的考克斯特-托德晶格。一个统一的建设给出的（甚至）类似物的Leech格，启发的事实，即（*）也是一组无平方阶的元素的Mathieu群M 23。
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 .