Let us define the set of binary matrices where the elements sum to $n$ as $\mathbb{A}(N,n) = \{A \in \{0,1\}^{N \times N} | \sum A_{i,j} = n\}$ . Given two binary matrices $A,B \in \mathbb{A}(N,n)$, and the set of all $N \times N$ permutation matrices as $\mathbb{E}$: Let us define $A \equiv B\iff\exists E\in \mathbb{E}:A=EBE$, i.e. A is some simultaneous permutation of the rows and columns of B.
I am trying to find a minimal subset of all binary matrices, $\mathbb{S} \subset \mathbb{A}(N,n)$ such that $\forall C \in \mathbb{A}(N,n), \exists! S \in \mathbb{S} :S\equiv C$, i.e. a reduction of symmetric matrices to a smaller subset.
Are there any simple, easily represented/computed sets that satisfy this property above?