Is This Collection Even Possible?

I got to thinking that it would be really cool to put together a set of binary bills that has all possible combinations of the two numbers. For example:00000001, 00000010, 00000011, 00000100, 00000101, 00000111, etc. I think there are 256 different combinations.

I have two options that I could go with:

A few months ago I found a strap of $1's that is 82222201-82222300. In there are these bills: 82222222, 82222228, and 82222282. So I could try to put together a set of 8's and 2's. I'd have three down, 253 to go.

I would prefer to put together a a "true binary" set of 0's and 1's.

I have a bill that I found a long time ago that is 00000001. I would think that this would be the 2nd hardest (or expensive) note to get. If it is even made, I think the hardest to get would be 00000000.

So I need to find out: Would it even be possible to put together this set? Is there even a serial number 00000000? I have also heard that the BEP always pulls out the "rollover" set that ends in 4 nines, and the next bill that ends in 4 zeros (e.g. 12349999 and 12350000). So these bills are never released.

Is it even possible to put together a set that has all combinations of 0's and 1's?