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Characteristic vectors of unimodular lattices which represent two. (English) Zbl 1216.11069

The author improves the known upper bound of the dimension \(n\) of an indecomposable unimodular lattice \(L\) whose shadow has the third largest possible length, \(s=n-16\) to \(n\leq 89\). G. Nebe and B. Venkov [J. Number Theory 99, No. 2, 307–317 (2003; Zbl 1081.11049)] proved that if \(L\) has no roots and \(s=n-16\), then \(\text{rank}(L)\leq 46\). This bound is attained by \(L=O_{23}\perp O_{23}\) where \(O_{23}\) is the shorter Leech lattice.

MSC:

11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11E12 Quadratic forms over global rings and fields

Citations:

Zbl 1081.11049
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References:

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