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 − 3 log2 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 − 3 log2 n consecutive words. Such Gray codes are ‘locally distance preserving’ in that Hamming distance equals index separation for nearby words in the sequence.