During my studies I recently met the problem of finding a binary variable $x \in \lbrace 0,1 \rbrace ^n$ that solves
$$Ax=b, \qquad \|x\|_1 \leq \delta$$
I am curious how this system can be solved. One idea I have is to consider it as the feasible region of an integer linear program and apply some of the strategies developed there. Therefore, I am currently searching for papers solving binary integer programs.
Does anyone know some good papers/books that treat this problem? Or different approaches that might work? And does it have a name under which I can search for it?