In designing reversible circuits one of the useful circuits is the decoder.
The operation of a decoder is naturally reversible, so it makes sense to be able to create one with no garbage outputs.
However, when I look up reversible decoder implementations for n greater than 2, I can't find any that reduce the garbage outputs to 0. Is there a reason for this? Has it been proven impossible, or has a solution just eluded research?
In particular this paper seems to show that many attempts to create 3-to-8 decoders seem to get stuck at 1 garbage output. I did notice that most papers seem to focus on general quantum cost, not just on garbage output. I suspect that if quantum cost is ignored it should be possible to create a 3-to-8 decoder with no garbage output lines.
I have tried with pen and paper to come up with such a design, to no avail, but haven't performed any kind of exhaustive search.