Characteristic vectors of unimodular lattices which represent two (Q933201)

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 \(n\leq 89\). \textit{G. Nebe} and \textit{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.