Definition of randomly generate in the parallelepiped I am reading the [CLT13] paper regarding multilinear maps over the integers and I have some troubles in understanding what the following random generation means:

The vectors of the matrix $r_i \in \mathbb{Z}^n$ are randomly and independently chosen from the half-open parallelepiped spanned by some vectors $\pi_1,...,\pi_n \in \mathbb{Z}^n$.

What I've understood: Those $r_i$ can be written as $r_i = a_1\pi_1+...+a_n\pi_n$ where each $a_i$ is random in $\mathbb{Z}$. Is it correct?
You might want to search for the word span to see exactly what the authors meant in the paper.
As @LeoDucas already commented, the answer is contained in Appendix E (page 22) of the same paper: