  This paper is about the rate of convergence of the Markov chain
$X_{n+1}=AX_{n}+B_{n}$ (mod $p$), where $A$ is an integer matrix with nonzero
eigenvalues and ${B_{n}}_{n}$ is a sequence of independent and identically
distributed integer vectors, with support not parallel to a proper subspace of
$Q^{k}$ invariant under $A$. If $|\lambda_{i}|\not=1$ for all eigenvalues
$\lambda_{i}$ of $A$, then $n=O((\ln p)^{2}) $ steps are sufficient and
$n=O(\ln p)$ steps are necessary to have $X_{n}$ sampling from a nearly uniform
distribution. Conversely, if $A$ has the eigenvalues $\lambda_{i}$ that are
roots of positive integer numbers, $|\lambda_{1}|=1$ and $|\lambda_{i}|>1$ for
all $i\not=1$, then $O(p^{2}) $ steps are necessary and sufficient.
