I have a lattice generated by $m$ integer vectors such that each generator lies in $\mathbb{R}^n$ for $m > n$. This clearly implies that these vectors are linearly dependent.
For lattices generated by linearly independent vectors, Minkowski's theorem implies an upper bound on the shortest vector, say in $\ell^\infty$ norm.
Is there an upper bound when the generators are linearly dependent?
If it helps, $m = \binom{n+1}{k+2}$ for some $k \in \mathbb{Z}$. Of particular interest to me is the case when $k = O\left(\sqrt{\frac{n}{\log(n)}}\right)$.