Consider the example where $n = 4$, bit 0 is fixed at 1, and bit 2 is fixed at 0. I would like to generate all $n$-bit numbers with those bits fixed. Essentially, everything I generate would have the form:
where $*$ represents a bit that can vary. The fixed bits can be in arbitrary positions, and there can be arbitrarily-many of them (anywhere from 1 bit fixed to $n-1$ bits fixed). Ideally, I would prefer these to be generated in lex order; what I'm looking to generate for the example case would be:
$$1000, 1001, 1100, 1101$$
I've only found solutions for doing something like this when specific bits are set to 1 - my question is more general, as it allows bits to be set to 0 or 1. What is a good algorithm (in terms of simplicity and efficiency) to do this for arbitrary $n$?