Suppose I have a Construction A lattice $$\mathcal{L} = \left\{\left[G |2\mathbb{I}_n\right]z\ :\ z\in\mathbb{Z}^{n+k} \right\}$$ where $G$ is a $n\times k$ matrix generating a binary linear code and $\mathbb{I}_n$ is the identity matrix, and suppose $u\in \mathbb{R}^n$ is fixed.
Is there a polynomial-time algorithm to find the vector $\hat{v}\in \mathcal{L}$ that has minimal, non-zero $|u\cdot \hat{v}|$?
That is, I want to solve $$ \begin{cases}\mathrm{minimize} \quad |u\cdot v| \\ \mathrm{subject\ to}\quad v\in \mathcal{L},\ |u\cdot v|>0 \end{cases}$$