Well-rounded lattice (awaiting review)

A lattice $L$ is called well-rounded if its set of minimal vectors $S(L)$ spans a sublattice $M$ of finite index, and $L$ is said to be generated by minimal vectors if $M = L$. The lattice $L$ is strongly well-rounded if the set $S(L)$ contains a basis for $L$. Being generated by minimal vectors is a strictly stronger condition than well-rounded for lattices of rank $\geq 5$, and a strictly weaker condition than strongly well-rounded for lattices of rank $\geq 10$.