Shrinking integer lattices. II (Q1192250)

[Part I (to appear).] The author considers sublattices of the integer lattice \(\mathbb{Z}^ k\) and asks for the maximal integer \(r\) such that there is a sublattice \(\Lambda'\subset\mathbb{Z}^ k\) isometric to \(r^{- 1/2}\Lambda\), where \(\Lambda\) is a given sublattice of rank \(k\) of \(\mathbb{Z}^ k\). The author gives the solution for \(k\leq 7\) and for \(k=8\) up to a factor 2.