How many number of classes over matrices with $\pm 1$ entries are there, where equivalence means row/column permutation and multiplication?

I'm interested in the number of inequivalent matrices with $\pm 1$ entries, where equivalence means row permutation, column permutation, row multiplication with $-1$, and column multiplication with $-1$?

If $\sim$ is the equivalence relation on $n \times m$ binary matrices such that $A \sim B$ iff one can obtain $B$ by applying any combination of the above operations (row/column permutation, row/column multiplication with $-1$) to $A$, I'm interested in the number of $\sim$-equivalence classes over all $n \times m$ matrices with $\pm 1$ entries. Thank you.

• When you write row/column permutation, you mean both at the same time like $P^{-1}AP$ or does a single operation suffice, like $AP$ (where $P$ is your permutation matrix)?$– SK19 May 6 '18 at 8:03 • Sorry for the ambiguity. I use a slash to save space. It means a single operation "row permutation or column permutation" (like$AP$, but not necessarily$P^{-1}AP$at the same time). The same applies for row/column multiplication. – Hang Wu May 6 '18 at 8:14 • I've updated the description in the problem. – Hang Wu May 6 '18 at 8:42 • Calculate it for some small values of$m$and$n\$, and then look it up in the Online Encyclopedia of Integer Sequences. – Gerry Myerson May 6 '18 at 10:35
• This paper shows some small results, but OEIS doesn't contain the sequence. – Hang Wu May 6 '18 at 10:37