By a *lattice* we mean sub-lattice of $\mathbb{Z}^n \cap V$, where $V$ is a subspace of $\mathbb{R}^n$ defined over $\mathbb{Q}$. We say that a lattice $\Lambda$ is *primitive* if a basis of $\Lambda$ can be extended to form a basis of $\mathbb{Z}^n$.

For a given lattice $\Lambda$, define its *orthogonal complement* $\Lambda^\ast$ by

$$\displaystyle \Lambda^\ast = \{\mathbf{x} \in \mathbb{Z}^n : \mathbf{x} \cdot \mathbf{y} = 0 \text{ for all } \mathbf{y} \in \Lambda\}.$$

It is known that if $\Lambda$ is primitive, then $\det(\Lambda) = \det(\Lambda^\ast)$.

Suppose that we are given a lattice $\Lambda$, contained in a proper subspace $V \subset \mathbb{Q}^n$. Suppose that we know have an explicit basis for $\Lambda^\ast$. Do we then have any control over the length of the shortest vector in $\Lambda$? Is there any way to explicitly control the length of the shortest basis vector of $\Lambda$ given data from $\Lambda^\ast$?