  The Diaconis--Gangolli random walk is an algorithm that generates an almost
uniform random graph with prescribed degrees. In this paper, we study the
mixing time of the Diaconis--Gangolli random walk restricted on $n\times n$
contingency tables over $\mathbb{Z}/q\mathbb{Z}$. We prove that the random walk
exhibits cutoff at $\frac{n^2}{4(1- \cos{\frac{2 \pi}{q}})} \log n, $ when
$\log q=o\left (\frac{\sqrt{\log n}}{\log \log n}\right )$.
