It is well known that given a $(n_s,m_t)$ configuration the following must hold:
$$ms=nt$$ $$s(t-1)+1\leq m$$ $$t(s-1)+1\leq n$$
However, for example, a $(43_7,43_7)$ configuration would be an order 6 projective plane but none such exists.
What is known about parameters that satisfy the above conditions but still do not produce a combinatorial configuration?
This paper covers what I was looking for:
https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-21/issue-1/The-Impossibility-of-Certain-Symmetrical-Balanced-Incomplete-Block-Designs/10.1214/aoms/1177729889.full