A few tuples at a time?

Question

I want to generate $n$ by $n$ binary arrays, and then reduce the output from those arrays by performing a number of permutations and eliminating duplicates. The problem is that the number of arrays increases as $2^{n^2}$, so storing every tuple in memory and comparing each permutation is impractical for $n>4$.

A working solution when $n<4$ is given here: Reduce the output from tuples by including symmetry?.

My questions is: how can I generate $n$ by $n$ binary arrays and eliminate duplicates (as defined by certain permutations) in a memory efficient manner?

For instance, is it possible to generate a few tuples at a time, run the permutations, check for duplicates, and then continue generating the remaining tuples?

P.S. I asked a different version of this question here: Constructing and Reducing Tuples in a Memory Efficient Manner, and have been unable to implement the solution given here: Memory efficient generation and selection of tuples.

Answer 1

I think you can do something with these identities:

Table[Partition[IntegerDigits[i, 2, 4], 2, 2], {i, 0, 2^4 - 1}] === Tuples[{0, 1}, {2, 2}]

Table[Partition[IntegerDigits[i, 2, 9], 3, 3], {i, 0, 2^9 - 1}] === Tuples[{0, 1}, {3, 3}]

So in general, you can say that this is True:

Table[Partition[IntegerDigits[i, 2, (n*n)], n, n], {i, 0, 2^(n*n) - 1}] === Tuples[{0, 1}, {n, n}]

(I checked this up to 5).

But this then gives you a way to just get any individual one:

f[n_, i_] := Partition[IntegerDigits[i, 2, (n*n)], n, n]

E.g.

f[10, 4938742893] //MatrixForm

Gives:

Hope this helps.