Random looking Gray Codes or Hamiltonian Cycles on Hypercubes

Question

Cyclic Gray codes come in many flavors and correspond 1-1 to Hamiltonian cycles on hypercubes. I would like to find a type that looks like a random walk on the hypercube. In a sense this is an oxymoron, since a real random walk will almost certainly not produce a Hamiltonian cycle. However, it might be possible to produce Gray codes where such that most segments up to some length look sufficiently random. Clearly the self avoiding property of a Hamiltonian path quickly becomes strong evidence of non randomness, but my critrea are not very strong I just need something where the structure is not completely obvious.