Oops! I found reference to chain codes in the Gray code wiki page, but didn't change the subject header. At 05:02 PM 3/28/2018, Henry Baker wrote:

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?