The title is admittedly noninformative but I could not figure out how to squeeze into it the description of the object I am interested in. Judge by yourself.

I have an alphabet on $d$ symbols. I want to build a rectangular array with $pd$ rows and $qd$ columns filled with them in such a way that each row contains $p$ copies of each symbol, each column contains $q$ copies of each symbol, and the following number $n$ is minimized.

For each pair $i,j$ of columns, let $n_{ij}$ be the number of rows where these columns contain identical symbols. Then $n$ is the largest of the $n_{ij}$ for all $i\ne j$.

I cannot even figure out what is the smallest possible value of $n$ for given $d,p,q$.

I tried to search for such structures. Not that I did not find anything. On the contrary, I found too many very similar investigations around gadgets called block designs, association schemes, Steiner triples and whatnot. Unfortunately I was unable to find the thing I need - I do not even give it a name since I am sure it already has a well-established name but I cannot figure out what it is.