Let $A=(a_{ij})$ be a $p\times q$ matrix with coefficients in $\mathbb{Z}$.
Is there a necessary and sufficient condition for $A$ to be a boundary matrix $\partial_n$ of some simplicial complex? (see for instance http://web.cse.ohio-state.edu/~wang.1016/courses/788/Lecs/lec7-qichao.pdf)
To give an example, say we have a 1-simplex $K=\{v_0,v_1,[v_0,v_1]\}$. Then the boundary matrix $\partial_1$ will be $\begin{pmatrix} -1\\1 \end{pmatrix}$.
Clearly, a necessary condition is that $A$ must have only entries in $\{-1,0,1\}$. Also necessary is that each column must have exactly $n+1$ non-zero entries.
However I am having some problem formulating a necessary and sufficient condition(s). It seems quite complicated, there ought to be even some conditions on $p$, $q$. For instance $p$ has to be greater than $q$.
Thanks for any help!