I would like to uniformly sample from all singular $n$ by $n$ Bernoulli matrices (that is each entry is $1$ or $0$ with probability $1/2$). I could of course just sample from all $n$ by $n$ Bernoulli matrices and reject those that are non-singular over $\mathbb{R}$ but for any moderate n that is extremely inefficient.

Is there an efficient way to do this?

Previously asked at http://stackoverflow.com/questions/21191831/how-to-uniformly-sample-from-singular-matrices where an answer over finite fields was given.

I would like to uniformly sample from all singular $n$ by $n$ Bernoulli matrices (that is each entry is $1$ or $0$ with probability $1/2$). I could of course just sample from all $n$ by $n$ Bernoulli matrices and reject those that are non-singular over $\mathbb{R}$ but for any moderate n that is extremely inefficient.

Is there an efficient way to do this?

Previously asked at https://stackoverflow.com/questions/21191831/how-to-uniformly-sample-from-singular-matrices where an answer over finite fields was given.