Happy New Year!

Suppose I would like to sample a $n \times n$ (0,1)-matrix whose trace is 0, and whose row sums and column sums are all $m$ with $1 \le m \le n-1.$ How can I sample this matrix uniformly?

Thank you very much!

Happy New Year!

Suppose I would like to sample a $n \times n$ (0,1)-matrix whose trace is 0, and whose row sums and column sums are all $m$ with $1 \le m \le n-1.$ How can I sample this matrix uniformly?

Thank you very much!

[Update based on the suggestion from user 44191] I am interested in the optimal way to sample such a matrix. What is the complexity of the problem as a function of $n$ and $m$? Thank you all!