Let $k<n$ be integers. Let $A\in \mathbb{Z}^{k \times n}$ be a sparse matrix, meaning that the number of nonzero entries in every row and every column is at most $O(1)$. Further, assume that there is an $O(1)$ bound on the absolute value of all nonzero entries. Let $\Lambda$ be the lattice $\{v \in \mathbb{Z}^n \mid Av=0\}$.
Does the sparsity of $A$ imply any useful bounds on the volume of the lattice $\Lambda$?
More strongly, does it imply the existence of a lattice basis for $\Lambda$ with better properties than one might expect by just applying the LLL algorithm? For example, can one give useful bounds on the norm of all vectors in the basis tighter than one would expect for an LLL-reduced basis?
