Finding the basis of the transpose of a q-ary lattice Given $q$ and a matrix $A \in \mathbb{Z}_q^{n \times m}$, the $q$-ary lattice is defined as 
$$\Lambda(A)=\{x \in \mathbb{Z}^m:Ax=0 \bmod q\} $$
An instance of a q-ary lattice and its short basis is computed in Generating short basis for hard random lattices. Once the short basis $T_A$ for $\Lambda(A)$ is given, computing the short vector $s$ in $\Lambda
(A)$ is given in SamplePre algorithm.

Is it possible to find a short basis for $\Lambda(A^T)$, if we are given a short basis of $ \Lambda(A)$? 

Basically I want to find the short vector $s^\prime$ such that $A^Ts^\prime=0 \bmod q$.
If $n&lt;m$ then for almost all matrices $A$ the columns of $A^T$ will be linearly independent and in that case $\Lambda(A^T)$ is the lattice generated by the basis  $qI$ (which is the shortest possible basis for this lattice).