Suppose $B \in \mathbb{R}^{n \times k}$ is a basis for a lattice $L$ of dimension $k$ ($n \geq k$ and the basis vectors are on the columns of $B$). Suppose also we have a set of $n$-vectors $\{ v_{(1)}, v_{(2)}, \cdots, v_{(\ell)} \}$, $1 \leq \ell < k$, which we know can be completed to a basis of $L$. Is there an efficient practical way to perform this completion?
As pointed out in the comments, the question is equivalent to: how does one practically complete an integer $n \times m$ matrix with $m < n$, to a unimodular $n \times n$ matrix (assuming the completion can be done)?