Binary Gray Codes with Long Bit Runs

We show that there exists an n-bit cyclic binary Gray code all of whose bit runs have length at least n - 3log₂ n. That is, there exists a cyclic ordering of \0,1\ⁿ such that adjacent words differ in exactly one (coordinate) bit, and such that no bit changes its value twice in any subsequence of n-3log₂ n consecutive words. Such Gray codes are 'locally distance preserving' in that Hamming distance equals index separation for nearby words in the sequence.