ON COUNTING SEQUENCES AND SOME RESEARCH QUESTIONS

A counting sequence is a list of all binary words of the same length. Counting sequences of any fixed odd Hamming distance between successive words (codewords) are discussed. Gray codes are examples of counting sequences having a single-bit change between neighboring codewords. We describe some results on Gray codes and highlight some research questions. The spectrum of bit changes or transition counts for individual variables for some uniform counting sequences is considered. We show some recent minor findings and pose remaining open questions