Let $\mathscr{M}[n,d]$ be collection of $n\times n$ symmetric matrices with real entries from $\{0,1\}$ such that every row/column is distinct with sum of every row/column being $d$.

What is minimum real rank of matrices in $\mathscr{M}[n,d]$?

Given real rank $r$, then $\mathscr{N}[r,d]$ be collection of symmetric rank $r$ square matrices with real entries from $\{0,1\}$ such that every row/column is distinct with sum of every row/column being $d$.

What is largest size of matrix in $\mathscr{N}[r,d]$?

Is there an algebraic or geometric way to describe these sets?