Most of the applications in the Wikipedia article on Gray codes consider *fixed length* codes for some length n. One of the properties of binary gray codes is that substrings of them contain all of the n-bit binary integers, but far more efficiently than simply concatenating all 2^n of them. I'm interested in a slightly different *infinite* sequence of bits, which contain these n-bit substrings as early as possible, s.t., for any n, there is a function f(n) that tells my how large of an initial substring of this infinite sequence I have to generate in order to make sure that all 2^n binary integers appear at least once. Furthermore, I may want to "tune" this infinite sequence in order to change the statistics of how frequently all of the different k-bit integers appear, how frequently all of the (k+1)-bit integers appear, etc. Has anyone studied such sequences?