In http://www.ijpam.eu/contents/2010-62-4/11/11.pdf I found the following statement (in my own words):
Let $H$ be an $(n-k)\times n$ matrix over $\mathbb{F}_p$ and let $A$ be an $(n-k)\times n$ matrix over $\mathbb{Z}$ such that $$H = A \otimes_{\mathbb{Z}} \mathbb{F}_p$$ Now, my question is: how do I interpret this statement? Should I view both $H$ and $A$ as vector spaces (given by the columns) and therefore as $\mathbb{Z}$-modules and $\mathbb{F}_p$ as a $\mathbb{Z}$-module induced by the canonical homomorphism? From this perspective, does the author mean that $H$ has the same entries as $A$, but considered as elements from $\mathbb{F}_p$ as opposed to integers?
Yeah.
More formally, $A$ is a map from $\Bbb Z^n$ to $\Bbb Z^{n-k}$, and $H=A\otimes \Bbb F_p$ really means $H=A\otimes \mathrm{Id}_{\Bbb F_p}$. So $H$ is a map from $\Bbb Z^n\otimes \Bbb F_p$ (which is $\Bbb F_p^n$) to $\Bbb Z^{n-r}\otimes \Bbb F_p$ (which is $\Bbb F_p^{n-r}$).