Lattices and codes with long shadows
Lattices and codes with long shadows
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带有长阴影的格子和代码
DOI：
10.4310/mrl.1995.v2.n5.a9
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
N. Elkies
中科院分区：
文献类型：
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作者：
N. Elkies
在之前的一篇论文（math.NT/9906019）中，我们证明了任何秩为n且与Z^n不等距的积分单模格L都有一个范数最多为n-8的特征向量。[L的“特征向量”是L中的向量w，使得L中所有v的2|(v,w-v)；已知特征向量的范数均等于n模8，并在L中构成一个2L的协集。这里我们使用模形式和秩<24的单模格的分类来找到最小特征向量范数为n-8的所有L。在此过程中，我们还得到了最小范数为2的非模格的同余和接吻数的下界。然后，我们陈述并证明了这些结果的自对偶码的类似结果，并通过“构造A”将它们直接与格问题联系起来。
In an earlier paper (math.NT/9906019) we showed that any integral unimodular lattice L of rank n which is not isometric with Z^n has a characteristic vector of norm at most n-8. [A "characteristic vector" of L is a vector w in L such that 2|(v,w-v) for all v in L; it is known that the characteristic vectors all have norm congruent to n mod 8 and comprise a coset of 2L in L.] Here we use modular forms and the classification of unimodular lattices of rank <24 to find all L whose minimal characteristic vectors have norm n-8. Along the way we also obtain congruences and a lower bound on the kissing number of unimodular lattices with minimal norm 2. We then state and prove analogues of these results for self-dual codes, and relate them directly to the lattice problems via "Construction A".